SADI    CARN  OT 

AT  THE  AGE  OP  17. 

(From  a  Portrait  by  Bailly.  1813.) 


REFLECTIONS 


ON  THE 


MOTIVE  POWER  OF  HEAT; 

FROM  THE  ORIGINAL  FRENCH  OF 

N.-L.-S.    CARNOT, 

Graduate  of  the  Polytechnic  School. 


ACCOMPANIED  BY 


AN  ACCOUNT  OF  CARNOT'S   THEORY. 

BY  SIR  WILLIAM  THOMSON  (LORD  KELVIN). 


EDITED  BY 


R.   H.   THURSTON,   M.A.,  LL.D.,  DR.ENG'G  ; 

Director  of  Sibley  College,  Cornell  University  ; 

"  Officier  de  VInstruction  Publique  de  France,"11 

etc.,  etc.,  etc. 


SECOND.  REVISED,   EDITIO1 
PIKST  THOUSAND. 


JOHN  WILEY   &  SONS, 


LONDON  :  CHAPMAN  &  HALL,  LIMITED. 


Copyright,  1890, 
ROBERT  H.  THURSTON. 


ROBERT  DRUMMOND,    ELECTROTVPER  AND   PRINTER,    NEW   YORK. 


DEDICATED 

TO 

SaDf  Garnet, 

PRESIDENT  OF  THE  FRENCH  REPUBLIC, 

THAT  DISTINGUISHED    MEMBER  OF  THE  PROFESSION  OF  ENGINEERING 

WHOSE  WHOLE  LlFE  HAS  BEEN  AN  HONOR  TO  HIS 

PROFESSION  AND  TO  HIS  COUNTRY  ; 

AND  WHO,    ELEVATED  TO  THE  HIGHEST  OFFICE  WITHIN  THE  GIFT  OF  THE 

FRENCH  NATION, 

HAS   PROVEN  BY  THE    QUIET  DIGNITY  AND  THE   EFFICIENCY  WITH  WHICH 
HE  HAS   PERFORMED   HIS  AUGUST  DUTIES  THAT  HE  IS 

A  WORTHY   MEMBER  OF  A  NOBLE  FAMILY, 
ALREADY  RENDERED   FAMOUS  BY  AN  EARLIER  SADI  CARNOT, 

NOW  IMMORTAL  IN  THE  ANNALS  OF  SCIENCE, 

AND  IS    HIMSELF  DESERVING  OF  ENROLMENT  IN  A  LIST  OF  GREAT  MEN 

WHICH  INCLUDES  THAT  OTHER  DISTINGUISHED   ENGINEER, 

OUR  OWN  FIRST  PRESIDENT, 

GEORGE   WASHINGTON. 


CONTENTS. 


i. 

PAGE 

THE  WORK  OF  N.-L.-SADi  CARNOT.    By  the  Editor,      I 

II. 

THE  LIFE  OF  N.-L.-SADi  CARNOT.  By  Mons.  H. 
Carnot,  .  .  .  .  .  .  . ;  .  .  .20 

III. 

REFLECTIONS  ON  THE  MOTIVE  POWER  OF  HEAT 
AND  ON  MACHINES  FITTED  TO  DEVELOP  THAT 
POWER.  By  Mons.  N.-L.-Sadi  Carnot,  .  .  37 

IV. 

ACCOUNT  OF  CARNOT'S  THEORY.  By  Sir  William 
Thomson  (Lord  Kelvin),  .  .  .  .  .127 

APPENDIX. 

A.  EXTRACTS  FROM    UNPUBLISHED  WRITINGS    OF 

CARNOT, '  .205 

B.  CARNOT'S  FOOT-NOTES, 237 

C.  NOTE  BY  THE  EDITOR, 261 

v 


PUBLISHERS'  NOTE. 

THE  raison  d'etre  of  the  following  translation  of 
the  famous  work  of  Carnot  is  not  the  usual  one, 
either  with  the  Publishers  or  the  Editor — expec- 
tation of  gain  in  either  purse  or  fame.  Neither 
could  reasonably  be  anticipated  from  the  reproduc- 
tion of  the  work  of  an  author  of  more  than  a  half- 
century  ago,  in  a  field  then  unrecognized,  and 
to-day  familiar  to  but  few  ;  and  especially  when, 
as  is  in  this  case  the  fact,  the  work  itself  has  been 
long  out  of  date  as  a  scientific  authority,  even  had 
it  ever  held  such  a  position.  It  could  not  be  pre- 
sumed that  a  very  large  proportion  of  even  the 
men  of  science  of  the  English-speaking  world 
would  be  sufficiently  familiar  with  the  subject,  or 
interested  in  its  origin,  to  purchase  such  a  relic  of 
a  primitive  period  as  is  this  little  book.  Nor 
could  the  translation  of  the  work,  or  the  gather- 
ing together  by  the  Editor  of  related  matter,  be 
supposed  likely  to  be  productive  of  any  form  of 
compensation.  The  book  is  published  as  matter 
of  limited  but  most  intense  scientific  interest, 
and  on  that  score  only. 

vii 


viii  PUBLISHERS'  NOTE. 

It  has  seemed  to  the  Editor  and  to  the  Pub- 
lishers that  the  product  of  the  wonderful  genius  of 
Carnot, — the  great  foundation-stone  of  one  of  the 
most  marvellous  and  important  of  modern  sciences, 
the  first  statement  of  the  grand  though  simple  laws 
of  Thermodynamics, — as  illustrated  in  this  one  lit- 
tle treatise,  should  be  made  accessible  to  all  who 
desire  to  study  the  work  in  English,  and  preserved, 
so  far  as  its  publication  in  this  form  could  ac- 
complish it,  as  a  permanent  memorial,  in  a  foreign 
tongue,  of  such  grand  truths,  and  of  such  a  great 
genius  as  was  their  discoverer.  It  is  with  this 
purpose  that  Publishers  and  Editor  have  co- 
operated in  this  project. 

The  book  consists,  as  will  be  seen  on  inspection, 
of  the  translation  of  Carnot's  Reflexions  sur  la 
Puissance  Motrice  du  Feu,  preceded  by  a  notice 
written  by  the  Editor  calling  attention  to  its 
remarkable  features,  and  its  extraordinary  char- 
acter as  the  product  of  a  most  remarkable  genius  ; 
and  by  a  biographical  sketch  of  the  great  author, 
written  by  his  brother,  Mons.  Hyppolyte  Carnot, 
which  sketch  we  find  in  the  French  copy  of  the 
work  as  published  by  Gauthier-Villars,  the  latest 
reproduction  of  the  book  in  the  original  toDgue. 
To  the  main  portion  of  the  book,  Carnot's  Re- 
flexions, is  appended  the  celebrated  paper  of  Sir 


PUBLISHERS'  NOTE.  ix 

William  Thomson,  his  "Account  of  Carnot's 
Theory,"  in  which  that  great  physicist  first  points 
out  to  the  world  the  treasure  so  long  concealed, 
unnoticed,  among  the  scientific  literature,  already 
mainly  antiquated,  of  the  first  quarter  of  the  nine- 
teenth century.  The  distinguished  writer  of  this 
paper  has  kindly  interested  himself  in  the  scheme 
of  the  Editor,  and  has  consented  to  its  insertion 
as  a  natural  and  desirable  commentary  upon  the 
older  work,  and  especially  as  exhibiting  the  rela- 
tions of  the  fundamental  principles  discovered  and 
enunciated  by  Carnot  to  the  modern  view  of  the 
nature  of  thermodynamic  phenomena — relations 
evidently  understood  by  that  writer,  but  not  by 
the  leaders  of  scientific  thought  of  his  time,  and 
therefore  ignored  by  him  in  the  construction  of 
his  new  science. 

The  Appendix  contains  a  number  of  Carnot's 
own  notes,  too  long  to  be  inserted  in  the  body  of 
the  paper  in  its  present  form,  and  which  have 
therefore  been  removed  to  their  present  location 
simply  as  a  matter  of  convenience  in  book- 
making. 

The  dedication  of  the  work  to  the  grand- 
nephew  of  the  author,  who  by  a  singular  coinci- 
dence happens  to-day  to  occupy  the  highest  posi- 
tion that  any  citizen  can  aspire  to  reach  in  that 


X  PUBLISHERS'  NOTE. 

now  prosperous  Kepublic,  will  be  recognized  as  in 
all  respects  appropriate  by  every  reader  of  the  work 
of  the  earlier  Sadi  Carnot  who  is  familiar  with 
the  character,  the  history,  the  attainments,  the 
achievements,  of  the  later  Sadi  Carnot  in  so 
many  and  widely  diverse  fields.  The  Carnot 
talent  and  the  Carnot  character  are  equally  ob- 
servable in  both  men,  widely  as  they  are  separated 
in  time  and  in  the  nature  of  their  professional 
labors.  Both  are  great  representatives  of  a  noble 
family,  whose  honor  and  fame  they  have  both 
splendidly  upheld. 

The  Publishers  offer  this  little  book  to  its 
readers  as  a  small,  yet  in  one  sense  not  unim- 
portant, contribution  to  the  great  cause  of  modern 
science,  as  a  relic,  a  memorial/a  corner-stone. 


NOTE  BY  THE  EDITOR. 

"  Je  me  suis  propose  de  grands  desseins  dans  ce 
petit  ouvrage"  as  Bernardin  de  Saint-Pierre  says  in 
the  preface  to  his  pathetic  story  of  Paulet  Virginie. 
I  have  sought  to  present  to  the  great  English- 
speaking  world  the  work  of  a  genius  hitherto  only 
known  to  a  few  men  of  science,  and  not  well  known, 
even  among  the  people  of  France,  for  whose  credit 
he  has  done  so  much.  In  placing  before  the  read- 
ers of  this  translation  his  book — small  of  size  but 
great  in  matter  as  it  is — I  feel  that  I  have  accom- 
plished an  easy  task,  but  one  of  real  importance. 
I  have  been  asked,  as  Corresponding  Member  for 
the  United  States  of  the  Societe  des  Ingenieurs 
Civils  de  France,  to  communicate  to  my  colleagues 
scientific  and  professional  memoirs  and  whatever 
may  be  of  interest  to  them — "en  un  mot,que  nous 
resserrions  les  liens  qui  font  des  ingenieurs  en  ge- 
neral une  seule  famille."  That  were  a  pleasant 
task;  but  a  grander  and  a  more  agreeable  one  still 
is  that  of  bringing  "  nearer  in  heart  and  thought " 
the  members  of  that  still  larger  community,  the 
men  of  science  of  the  world,  and  of  weaving  still 

xi 


xii  NOTE  BY  THE  EDITOR. 

more  firmly  and  closely  those  bonds  of  kindly 
thought  and  feeling  which  are  growing  continually 
more  numerous  and  stronger  as  the  nations  are 
brought  to  see  that  humanity  is  larger  and  more 
important  than  political  divisions,  and  that  the 
labors  of  educated  men  and  of  the  guiding  minds 
in  the  great  industries  are  constantly  doing  more 
to  promote  a  true  brotherhood  of  mankind  inan 
ever  have,  or  ever  can,  the  greatest  statesmen. 

When  the  wonderful  intellectual  accomplish-- 
menis  of  men  like  the  elder  Sadi  Carnot  become 
known  and  appreciated  by  the  world,  much  more 
will  have  been  accomplished  in  tliis  direction.  It 
is  perhaps  from  this  point  of  view  that  the  impor- 
tance of  such  work  will  be  most  fully  recognized. 
When  the  little  treatise  which  is  here  for  the  first 
time  published  in  English  becomes  familiar  to 
those  for  whom  it  is  intended,  it  will  be,  to  many 
at  least,  a  matter  of  surprise  no  less  than  pleasure 
to  discover  that  France  has  produced  a  writer  on 
this  now  familiar  subject  whose  inspiration  antici- 
pated many  of  the  principles  that  those  founders 
of  the  modern  science,  Eankine  and  Clausius, 
worked  out  through  the  tedious  and  difficult 
methods  of  the  higher  mathematics,  and  which 
were  hailed  by  their  contemporaries  as  marvellous 
discoveries. 


NOTE  TO  SECOND  EDITION. 


THE  present  edition  of  this  little  work  is  im- 
proved by  the  removal  of  a  few  errata  observed  in 
the  first  issue,  and  by  the  addition  of  a  recent  and 
excellent  portrait  of  Lord  Kelvin,  as  a  frontispiece 
to  his  era-making  paper,  at  page  127.  This  pic- 
ture, taken  within  the  last  year,  is  thought  by  the 
friends  of  its  distinguished  subject  to  be  one  of  the 
best  yet  produced.  That  it  is  satisfactory  to  him 
and  his  friends  is  indicated  by  the  fact  that  the 
original  of  this  reproduction  was  presented  to  the 
writer  by  Lady  Kelvin,  in  1895,  immediately  after 
it  was  taken,  and  the  autograph  supplied  by  her 
distinguished  husband.  The  Editor  takes  this 
occasion  to  acknowledge  cordially  the  letters  of 
appreciation  and  commendation  received  from 
those  who  have  agreed  with  M.  Haton  de  la  Gou- 
pilliere  that  the  translation  of  Carnot  and  its 
publication  in  this  manner,  with  the  famous  paper 
of  Lord  Kelvin,  will  be  considered  as  worthy  of 
approval  by  English-speaking  readers  as  well  as 
"appreciated  by  the  whole  French  nation." 

xiii 


THE  WORK  OF  SADI  CARNOT, 
BY  THE  EDITOR. 

NICOLAS-LEONAKD-SADI  CAK:NX)T  was,  perhaps, 
the  greatest  genius,  in  the  department  of  physical 
science  at  least,  that  this  century  has  produced. 
By  this  I  mean  that  he  possessed  in  highest  degree 
that  combination  of  the  imaginative  faculty  with 
intellectual  acute  ness,  great  logical  power  and  ca- 
pacity for  learning,  classifying  and  organizing  in 
their  proper  relations,  all  the  facts,  phenomena, 
and  laws  of  natural  science  which  distinguishes 
the  real  genius  from  other  men  and  even  from  the 
simply  talented  man.  Only  now  and  then,  in  the 
centuries,  does  such  a  genius  come  into  view. 
Euclid  was  such  in  mathematics ;  Newton  was 
such  in  mechanics  ;  Bacon  and  Compte  were  such 
in  logic  and  philosophy  ;  Lavoisier  and  Davy  were 
such  in  chemistry;  and  Fourier,  Thomson,  Max- 

1 


2  THE  WORK  OF  SAD1  CAENOT. 

well,  and  Clausius  were  such  in  mathematical 
physics.  Among  engineers,  we  have  the  exam- 
ples of  Watt  as  inventor  and  philosopher,  Eankine 
as  his  mathematical  complement,  developing  the 
theory  of  that  art  of  which  Watt  illustrated  the 
practical  side  ;  we  have  Hirn  as  engineer- experi- 
mentalist, and  philosopher,  as  well ;  Corliss  as  in- 
ventor and  constructor ;  and  a  dozen  creators  of 
the  machinery  of  the  textile  manufactures,  in 
which,  in  the  adjustment  of  cam-work,  the  high- 
est genius  of  the  mechanic  appears. 

But  Carnot  exhibited  that  most  marked  charac- 
teristic of  real  genius,  the  power  of  applying  such 
qualities  as  I  have  just  enumerated  to  great  pur- 
poses and  with  great  result  while  still  a  youth. 
Genius  is  not  dependent,  as  is  talent,  upon  the 
ripening  and  the  growth  of  years  for  its  pres- 
cience ;  it  is  ready  at  the  earliest  maturity,  and 
sometimes  earlier,  to  exhibit  its  marvellous  works  ; 
as,  for  example,  note  Hamilton  the  mathema- 
tician and  Mill  the  logician ;  the  one  becoming 
master  of  a  dozen  languages  when  hardly  more 
than  as  many  years  of  age,  reading  Newton's  Prin- 
cipia  at  sixteen  and  conceiving  that  wonderful 
system,  quaternions,  at  eighteen  ;  the  other  com- 
petent to  begin  the  study  of  Greek  at  three,  learn- 
ing Latin  at  seven  and  reading  Plato  before  he 


UNIVERSITY 


THE   WORK  OF  8ADI  CARNOT. 

was  eight.  Carnot  had  done  his  grandest  work  of 
the  century  in  his  province  of  thought,  and  had 
passed  into  the  Unseen,  at  thirty-six  ;  his  one  little 
volume,  which  has  made  him  immortal,  was  writ- 
ten when  he  was  but  twenty-three  or  twenty-four. 
It  is  unnecessary,  here,  to  enter  into  the  particu- 
lars of  his  life  ;  that  has  been  given  us  in  ample 
detail  in  the  admirable  sketch  by  his  brother 
which  is  here  republished.  It  will  be  quite  suf- 
ficient to  indicate,  in  a  few  words,  what  were  the 
conditions  amid  which  he  lived  and  the  relation 
of  his  work  to  that  great  science  of  which  it  was 
the  first  exposition. 

At  the  time  of  Carnot,  the  opinion  of  the 
scientific  world  was  divided,  as  it  had  been  for 
centuries,  on  the  question  of  the  true  nature  of 
heat  and  light,  and  as  it  still  is,  to  a  certain  ex- 
tent, regarding  electricity.  On  the  one  hand  it 
was  held  by  the  best-known  physicists  that  heat 
is  a  substance  which  pervades  all  bodies  in  greater 
or  less  amount,  and  that  heating  and  cooling  are 
simply  the  absorption  and  the  rejection  of  this 
"  imponderable  substance  "  by  the  body  affected  ; 
while,  on  the  other  hand,  it  was  asserted  by  a 
small  but  increasing  number  that  heat  is  a 
"mode  of  motion,"  a  form  of  energy,  not  only 
imponderable,  but  actually  immaterial  ;  a  quality 


4  THE  WORK  OF  SADI  CARNOT. 

of  bodies,  not  a  substance,  and  that  it  is  identical, 
in  its  nature,  with  other  forms  of  recognizable 
energy,  as,  for  example,  mechanical  energy.  A 
quarter  of  a  century  before  Carnot  wrote,  the  ex- 
periments of  Rumford  and  of  Davy  had  been  cru- 
cial in  the  settlement  of  the  question  and  in  the 
proof  of  the  correctness  of  the  second  of  the  two 
opposing  parties  ;  but  their  work  had  not  become 
so  generally  known  or  so  fully  accepted  as  to  be 
acknowledged  as  representative  of  the  right  views 
of  the  subject.  The  prevalent  opinion,  following 
Newton,  was  favorable  to  the  first  hypothesis ; 
and  it  was  in  deference  to  this  opinion  that  Carnot 
based  his  work  on  an  inaccurate  hypothesis ; 
though,  fortunately,  the  fact  did  not  seriously 
militate  against  its  value  or  his  credit  and  fame. 

"With  true  philosophical  caution,  he  avoids 
committing  himself  to  this  hypothesis  ;  though  he 
makes  it  the  foundation  of  his  attempt  to  discover 
how  work  is  produced  from  heat."  * 

The  results  of  Carnot's  reasoning  are,  fortu- 
nately, mainly  independent  of  any  hypothesis  as 
to  the  nature  of  heat  or  the  method  or  mechanism 
of  development  and  transfer  or  transformation  of 
its  energy.  Carnot  was  in  error  in  assuming  no 

*  Tait :  Thermodynamics,  p.  13. 


THE  WORK  OF  8ADI  CARNOT.  5 

loss  of  heat  in  a  completed  cycle  and  in  thus  ignor- 
ing the  permanent  transformation  of  a  definite 
proportion  into  mechanical  energy  ;  but  his  propo- 
sition that  efficiency  increases  with  increase  of 
temperature-range  is  still  correct ;  as  is  his  asser- 
tion of  its  independence  of  the  nature  of  the 
working  substance. 

Oarnot's  "Reflexions  sur  la  Puissance  Motrice 
du  Feu"  published  in  1824,  escaped  notice  at  the 
time,  was  only  now  and  then  slightly  referred  to 
later,  until  Clapeyron  seized  upon  its  salient  ideas 
and  illustrated  them  by  the  use  of  the  Watt  dia- 
gram of  energy,  and  might,  perhaps,  have  still  re- 
mained unknown  to  the  world  except  for  the  fact 
that  Sir  William  Thomson,  that  greatest  of  modern 
mathematical  physicists,  fortunately,  when  still  a 
youth  and  at  the  commencement  of  his  own  great 
work,  discovered  it,  revealed  its  extraordinary 
merit,  and,  readjusting  Carnot's  principles  in  ac- 
cordance with  the  modern  views  of  heat-energy, 
gave  it  the  place  that  it  is  so  well  entitled  to  in 
the  list  of  the  era-making  books  of  the  age.  But 
it  still  remained  inaccessible  to  all  who  could  not 
find  the  original  paper  until,  only  a  few  years 
since,  it  was  reprinted  by  Gauthier-Villars,  the 
great  publishing  house  of  Paris,  accompanied  by  a 
biographical  sketch  by  the  younger  brother,  which 


6  THE  WORK  OF  SADI  CARNOT. 

it  has  been  thought  wise  to  reproduce  with  the 
translation  of  Carnot's  book.  In  making  the 
translation,  also,  this  later  text  has  been  -followed  ; 
and  now,  for  the  first  time,  so  far  as  is  known  to 
the  writer,  the  work  of  Carnot  is  made  accessible 
to  the  reader  in  English. 

The  original  manuscript  of  Carnot  has  been  de- 
posited by  his  brother  in  the  archives  of  the 
French  Academy  of  Sciences,  and  thus  insured 
perpetual  care.  The  work  of  Carnot  includes  not 
only  the  treatise  which  it  is  the  principal  object  of 
this  translation  to  give  to  our  readers,  but  also  a 
considerable  amount  of  hitherto  unpublished  mat- 
ter which  has  been  printed  by  his  brother,  with 
the  new  edition  of  the  book,  as  illustrative  of  the 
breadth  and  acuteness  of  the  mind  of  the  Founder 
of  the  Science  of  Thermodynamics. 

These  previously  unpublished  materials  consist 
of  memoranda  relating  to  the  specific  heats  of 
substances,  their  variations,  and  various  other 
facts  and  data,  and  principles  as  well ;  some  of 
which  are  now  recognized  as  essential  elements  of 
the  new  science,  even  of  its  fundamental  part. 
The  book  is  particularly  rich  in  what  have  been 
generally  supposed  to  be  the  discoveries  of  later 
writers,  and  in  enunciations  of  principles  now 
tecognized  as  those  forming  the  base  and  the  sup- 


THE  WORK  OF  SADI  CARNOT.  7 

porting  framework  of  that  latest  of  the  sciences. 
As  stated  by  Tait,  in  his  history  of  Thermody- 
namics, the  "  two  grand  things"  which  Carnot  ori- 
ginated and  introduced  were  his  idea  of  a  "cycle" 
and  the  notion  of  its  "  reversibility,"  when  perfect. 
"  Without  this  work  of  Carnot,  the  modern  theory 
of  energy,  and  especially  that  branch  of  it  which 
is  at  present  by  far  the  most  important  in  prac- 
tice, the  dynamical  theory  of  heat,  could  not  have 
attained  its  now  enormous  development."  These 
conceptions,  original  with  our  author,  have  been, 
in  the  hands  of  his  successors,  Clausius  and  other 
Continental  writers,  particularly,  most  fruitful  of 
interesting  and  important  results  ;  and  Clapeyron's 
happy  thought  of  so  employing  the  Watt  diagram 
of  energy  as  to  render  them  easy  of  comprehen- 
sion has  proved  a  valuable  aid  in  this  direction. 

The  exact  experimental  data  needed  for  numer- 
ical computations  in  application  of  Carnot's  prin- 
ciples were  inaccessible  at  the  date  of  his  writing; 
they  were  supplied,  later,  by  Mayer,  by  Cold  ing, 
by  Joule,  and  by  later  investigators.  Even  the 
idea  of  equivalence,  according  to  Hypolyte  Car- 
not, was  not  originally  familiar  to  the  author  of 
this  remarkable  work;  but  was  gradually  developed 
and  defined  as  he  progressed  with  his  philosophy. 
It  is  sufficiently  distinctly  enunciated  in  his  later 


8  THE  WORK  OF  8ADI  CARNOT. 

writings.  He  then  showed  a  familiarity  with 
those  notions  which  have  been  ascribed  generally 
to  Mayer  and  which  made  the  latter  famous,  and 
with  those  ideas  which  are  now  usually  attributed 
to  Joule  with  similar  result.  He  seems  actually  to 
have  planned  the  very  kind  of  research  which  Joule 
finally  carried  out.  All  these  advanced  views 
must,  of  course,  have  been  developed  by  Carnot 
before  1832,  the  date  of  his  illness  and  death,  and 
ten  or  fifteen  years  earlier  than  they  were  made 
public  by  those  who  have  since  been  commonly 
considered  their  discoverers.  These  until  lately 
unpublished  notes  of  Carnot  contain  equally  well- 
constructed  arguments  in  favor  of  the  now  accepted 
theory  of  heat  as  energy.  While  submitting  to 
the  authority  of  the  greatest  physicists  of  his  time, 
and  so  far  as  to  make  their  view  the  basis  of  his 
work,  to  a  certain  extent,  he  nevertheless  adhered 
privately  to  the  true  idea.  His  idea  of  the  equiva- 
lence of  heat  and  other  forms  of  energy  was  as  dis- 
tinct and  exact  as  was  his  notion  of  the  nature  of 
that  phenomenon.  He  states  it  with  perfect  ac- 
curacy. 

In  making  his  measures  of  heat-energy,  he  as- 
sumes as  a  unit  a  measure  not  now  common,  but 
one  which  may  be  easily  and  conveniently  reduced 
to  the  now  general  system  of  measurement.  He 


THE  WORK  OF  SADI  CARNOT.  9 

takes  the  amount  of  power  required  to  exert  an 
energy  equal  to  that  needed  to  raise  one  cubic 
meter  of  water  through  a  height  of  one  meter, 
as  his  unit;  this  is  1000  kilogrammeters,  taken 
as  his  unit  of  motive  power;  while  he  says  that 
this  is  the  equivalent  of  2.7  of  his  units  of 
heat;  which  latter  quantity  would  be  destroyed 
in  its  production  of  this  amount  of  power,  or 
rather  work.  His  unit  of  heat  is  thus  seen 
to  be  1 000  -f-  2. 7,  or  370  kilogram  meters.  This 
is  almost  identical  with  the  figure  obtained  by 
Mayer,  more  than  ten  years  later,  and  from 
presumably  the  same  approximate  physical  data, 
the  best  then  available,  in  the  absence  of  a  Reg- 
nault  to  determine  the  exact  values.  Mayer  ob- 
tained 365,  a  number  which  the  later  work  of 
Regnault  enabled  us  to  prove  to  be  15  per  cent, 
too  low,  a  conclusion  verified  experimentally  by 
the  labors  of  Joule  and  his  successors.  Carnot  was 
thus  a  discoverer  of  the  equivalence  of  the  units  of 
heat  and  work,  as  well  as  the  revealer  of  the  prin- 
ciples which  have  come  to  be  known  by  his  name. 
Had  he  lived  a  little  longer,  there  can  be  little 
doubt  that  he  would  have  established  the  facts,  as 
well  as  the  principles,  by  convincing  proof.  His 
early  death  frustrated  his  designs,  and  deprived  the 


10  THE  WORK  OF  SADI  CARROT. 

world  of  one  of  its  noblest  intellects,  just  when  it 
was  beginning  its  marvellous  career. 

The  following  sentence  from  Carnot  illustrates 
in  brief  his  wonderful  prescience;  one  can  hardly 
believe  it  possible  that  it  should  have  been  written 
in  the  first  quarter  of  the  nineteenth  century: 
"  On  pent  done  poser  en  these  generate  que  la  puis- 
sance motrice  est  en  quantite  invariable  dans  la 
Nature;  qu' elle  n'est  jamais,  a  proprement  parler, 
ni  produite,  ni  detruite.  A  la  verite,  elle  change 
de  forme)  c'est  a  direqu' elle  produittantotun  genre 
de  mouvement,  tantot  un  autre;  mais  elle  n'est 
jamais  aneantie."  It  is  this  man  who  has  prob- 
ably inaugurated  the  development  of  the  modern 
science  of  thermodynamics  and  the  whole  range  of 
sciences  dependent  upon  it,  and  who  has  thus  made 
it  possible  to  construct  a  science  of  the  energetics 
of  the  universe,  and  to  read  the  mysteries  of  every 
physical  phenomenon  of  nature;  it  is  this  man  who 
has  done  more  than  any  contemporary  in  his  field, 
and  who  thus  displayed  a  more  brilliant  genius 
than  any  man  of  science  of  the  nineteenth  century: 
yet  not  even  his  name  appears  in  the  biographical 
dictionaries;  and  in  the  Encyclopaedia  Britannica 
it  is  only  to  be  found  incidentally  in  the  article  on 
Thermodynamics. 

Throughout  his  little  book,  we  find   numerous 


THE  WORK  OF  SADI  CARNOT.  11 

proofs  of  his  clearness  of  view  and  of  the  wonder- 
ful powers  of  mind  possessed  by  him.  He  opens 
his  treatise  by  asserting  that  "  C'est  a  la  chaleur 
que  doivent  etre  attribute  les  grands  mouvements 
qui  frappent  nos  regards  sur  la  terre;  c'est  a  elle 
que  sont  dues  les  agitations  de  V atmosphere,  Vas- 
cension  des  nuages,  la  chute  des  pluies  et  ties  autres 
meteores,  les  courants  d'eau  qui  sillonnent  la  surface 
du  globe  et  dont  Vhomme  est  parvenue  a  employer 
pour  son  usage  une  faible  partie;  en  fin  les  tremble- 
menfs  de  terre,  les  eruptions  volcaniques  reconnais- 
sent  aussi  pour  cause  la  chaleur" 

Carnot  was  the  first  to  declare  that  the  maximum 
of  work  done  by  heat,  in  any  given  case  of  appli- 
cation of  the  heat-energy,  is  determined  solely  by 
the  range  of  temperature  through  which  it  fell  in 
the  operation,  and  is  entirely  independent  of  the 
nature  of  the  working  substance  chosen  as  the 
medium  of  transfer  of  energy  and  the  vehicle  of 
the  heat.  His  assumption  of  the  materiality  of 
heat  led,  logically,  to  the  conclusion  that  the 
same  quantity  of  heat  was  finally  stored  in  the 
refrigerator  as  had,  initially,  left  the  furnace,  and 
that  the  effect  produced  was  a  consequence  of  a  fall 
of  temperature  analogous  to  a  fall  of  water;  but, 
aside  from  this  error — which  he  himself  was  evi- 


12  THE  WORK  OF  8ADI  CARNOT. 

dently  inclined  to  regard  as  such, — his  process  and 
argument  are  perfectly  correct.* 

Throughout  his  whole  work  are  distributed  con- 
densed assertions  of  principles  now  well  recognized 
and  fully  established,  which  indicate  that  he  not 
only  had  anticipated  later  writers  in  their  estab- 
lishment, but  that  he  fully  understood  their  real 
importance  in  a  theory  of  heat-energy  and  of  heat- 
engines.  In  fact,  he  often  italicizes  them,  placing 
them  as  independent  paragraphs  to  more  thor- 
oughly impress  the  reader  with  their  fundamental 
importance.  Thus  he  says  :  "  Partout  ou  il  existe 
une  difference  de  temperature,  il  pent  y  avoir  pro- 
duction de  puissance  motrice;"  and  again,  this 
extraordinary  anticipation  of  modern  science  :  ' '  le 
maximum  de  puissance  resultant  de  I'emploi  de  la 
vapeur  est  aussi  le  maximum  de  puissance  motrice 
realisable  par  quelque  moyen  que  ce  soit." 

(( La  puissance  motrice  de  la  chaleur  est  inde- 
pendante  des  agents  mis  en  ceuvre  pour  la  realiser  ; 
sa  quant  ite  est  fixee  uniquement  par  les  temper  a- 


*  Account  of  Carnot's  Theory  of  the  Motive  Power  of 
Heat;  Sir  Wm.  Thomson;  Trans.  Roy.  Soc.  of  Edin- 
burgh, xvi.  1849;  and  Math,  and  Phys.  Papers,  xli.  vol.  1 
(Cambridge,  1882),  p.  113.  In  this  paper  the  corrections  due 
to  the  introduction  of  the  dynamic  theory  are  first  applied. 


THE  WORK  OF  SADI  CARNOT.  13 

tures  des  corps  entre  lesquels  sefait,  en  dernier  re- 
sultat,  le  transport  du  calorique." 

"  Lorsqu'un  gaz  passe,  sans  changer  de  tempera- 
ture, d'un  volume  et  d'une  pression  determines  a  une 
autre  pression  egalement  determinee,  la  quantite 
de  calwique  absorbee  ou  abandon?iee  est  toujours  la 
meme,  quelle  que  soit  la  nature  du  gaz  choisi  comme 
sujet  a"  experience." 

Perhaps  as  remarkable  a  discovery  as  any  one  of 
the  preceding  (and  one  which,  like  those,  has  been 
rediscovered  and  confirmed  by  later  physicists  ; 
one  which  was  the  subject  of  dispute  between 
Clausius,  who  proved  its  truth  by  the  later  methods 
which  are  now  the  source  of  his  fame,  and  the 
physicists  of  his  earlier  days,  who  had  obtained 
inaccurate  measures  of  the  specific  heats  of  the 
gases; — values  which  were  finally  corrected  by  Reg- 
nault,  thus  proving  Carnot  and  Clausius  to  be 
right — is  thus  stated  by  Carnot,  and  is  italicized 
in  his  manuscript  and  book  : 

"  La  difference  entre  la  clialeur  specijique  sous 
pression  constante  et  la  clialeur  specijique  sous  vo- 
lume constant  est  la  meme  pour  tous  les  gaz." 

He  bases  his  conclusion  upon  the  simplest  of 
thermodynamic  considerations.  He  says  that  the 
increase  of  volumes  with  the  same  differences  of 
temperature  are  the  same,  according  to  Gay-Lussao 


14     THE  WORK  OF  SADI  CAENOT. 

and  Dal  ton  ;  and  that,  therefore,  according  to  the 
laws  of  thermodynamics  as  lie  has  demonstrated 
them,  the  heat  absorbed  with  equal  augmenta- 
tions of  volume  being  the  same,  the  two  specific 
heats  are  constant,  and  their  difference  as  well.  As 
will  be  seen  on  referring  to  the  text,  he  bases  upon 
this  principle  a  determination  of  the  specific  heats 
of  constant  volume,  taking  as  his  values  of  the  de- 
termined specific  heats  of  constant  pressure  those 
of  Delaroche  and  Berard,  making  the  constant 
difference  0.300,  that  of  air  at  constant  pressure 
being  taken  as  the  standard  and  as  unity.  The 
establishment  of  this  point,  in  the  face  of  the  op- 
position, and  apparently  of  the  facts,  of  the  best 
physicists  of  his  time,  was  one  of  those  circum- 
stances which  did  so  much  to  win  for  Clausius  his 
great  fame.  How  much  greater  credit,  then, 
should  be  given  Carnot,  who  not  only  anticipated 
the  later  physicists  in  this  matter,  but  who  must 
have  enunciated  his  principle  under  far  more  seri- 
ous discouragements  and  uncertainty  ! 

It  must  be  remembered,  when  reading  Oar- 
not,  that  ,all  the  "constants  of  nature"  were,  in 
his  time,  very  inaccurately  ascertained.  It  is  only 
since  the  time  of  Regnault's  grand  work  that  it  has 
been  the  rule  that  such  determinations  have  been 
published  only  when  very  exactly  determined.  No 


THE   WORK  OF  SADI  CARNOT.  15 

change  has  been  attempted  in  Carnot's  figures,  in 
any  respect ;  as  it  would  be  far  less  satisfactory  to 
read  a  paraphrased  work,  and  the  exact  figures  are 
now  easily  accessible  to  every  one,  and  his  compu- 
tations may  all  be  made,  if  desired,  on  the  basis  of 
modern  data.  Sir  William  Thomson  has  already 
performed  this  task  in  the  paper  appended. 

Throughout  the  whole  of  this  treatise,  small  as 
it  is,  we  find  distributed  a  singular  number  of 
these  anticipations  of  modern  thermodynamic 
principles.  Studying  the  relation  of  heat-energy 
to  work  done,  he  concludes  : 

"La  chute  du  calorique  produit  plus  de  puis- 
sance motrice  dans  les  degres  inferieurs  que  dans 
Us  degres  superieurs." 

We  to-day  admit  that,  since  the  one  degree  at  a 
low  temperature,  and  the  corresponding  quantity 
ct  heat,  are  larger  fractions  of  the  total  tempera- 
ture, and  the  total  heat  stored  in  the  substance, 
than  the  one  degree  at  a  higher  point  on  the  scale 
of  absolute  temperature,  this  principle  of  Carnot 
has  become  obvious. 

In  the  enunciation  of  the  essential  principles  of 
efficiency  of  the  heat-engine,  we  find  the  proofs  of 
this  same  wonderful  prescience.  He  asserts  that, 
for  best  effect  :  "  (1)  The  temperature  of  the 
working  fluid  must  be  raised  to  the  highest  degree 


16     THE  WORK  OF  SADI  CARNOT. 

possible,  in  order  to  secure  a  commensurate  range 
of  temperature  ;  (2)  The  cooling  must  be  carried 
to  the  lowest  point  on  the  scale  that  may  be  found 
practicable  ;  (3)  The  passage  of  the  fluid  from  the 
upper  to  the  lower  limit  of  temperature  must  be 
produced  by  expansion;"  i.e.,  "it  is  necessary 
that  the  cooling  of  the  gas  shall  occur  sponta- 
neously by  its  rarefaction  ;"  which  is  simply  his 
method  of  stating  the  now  universally  understood 
principle  that,  for  highest  efficiency,  the  expansion 
must  be  adiabatic,  from  a  maximum  to  a  mini- 
mum temperature.  He  goes  on  to  explain  these 
principles,  and  then  says  that  the  advantage  of 
high-pressure  engines  lies  "  essentiellement  dans  la 
faculte  de  rendre  utile  vne  plus  grande  chute  de  ca- 
loriqne."  This  principle,  as  a  practical  system  of 
operation,  had  already,  as  he  tells  us,  been  enunci- 
ated by  M.  Clement,  and  had  been  practised,  as 
we  well  know,  since  the  days  of  its  originator, 
Watt  ;  but  Carnot  saw  clearly  the  thermodynamic 
principle  which  underlies  it,  and  as  clearly  states 
it,  for  the  first  time. 

He  sees  clearly,  too,  the  reasons  for  the  attempts 
of  Hornblower  and  of  Woolf,  premature  as  they 
proved  and  as  he  also  sees,  in  the  introduction  of 
the  compound  engine,  and  even  suggests  that  this 
idea  might  be  still  further  developed  by  the  use 


THE  WORK  OF  SADI  CARNOT,  17 

of  a  triple-expansion  engine,  a  type  which  is  to- 
day just  coming  into  use,  more  than  a  half -cen- 
tury after  Carnot/s  date.  He  recognizes  the  ad- 
vantages of  the  compound  engine  in  better  distri- 
bution of  pressures  and  in  distribution  of  the  work 
of  expansion,  but  does  not,  of  course,  perceive  the 
then  undiscovered  limitation  of  the  efficiency  of 
the  simple  engine,  due  to  ' ' cylinder  condensation," 
which  has  finally  led,  perhaps  more  than  any  other 
circumstance,  to  its  displacement  so  largely  by 
the  multi-cylinder  machine.  No  one  has  more  ex- 
actly and  plainly  stated  the  respective  advantages 
to  be  claimed  for  air  and  the  gases,  used  as  work- 
ing fluids  in  heat-engines,  than  does  Carnot  ;  nor 
does  any  one  to-day  better  recognize  the  difficul- 
ties which  lie  in  the  path  to  success  in  that  direc- 
tion, in  the  necessity  of  finding  a  means  of  hand- 
ling them  at  high  temperatures  and  of  securing 
high  mean  pressures. 

His  closing  paragraph  shows  his  extraordinary 
foresight,  and  the  precision  with  which  that  won- 
derful intellect  detected  the  practical  elements 
of  the  problem  which  the  engineer,  from  the  days 
of  Savery,  of  Newcomen,  and  of  Watt,  has  been 
called  upon  to  study,  and  the  importance  of  the 
work,  which  he  began,  in  the  development  of  a 
theory  of  the  action,  or  of  the  operation,  of  the 


18  THE  WORK  OF  SADI  CARNOT. 

heat-engines,  which  should  give  effective  assistance 
in  the  development  of  their  improved  forms : 

"  On  ne  doit  pas  se  flatter  de  mettre  jamais  a 
profit,  dans  la  pratique,  toute  la  puissance  des  com- 
bustibles. Les  tentatives  que  Von  ferait  pour  ap- 
procher  ce  resultat  seraient  meme  plus  nuisiW.es 
qu'utiles,  si  ellex  faisaient  negliger  d'autres  conside- 
rations importantes.  L'economie  du  combustible 
n'est  qu'une  des  conditions  a  remplir  par  les  ma- 
chines a  feu;  dans  beaucoup  de  cir  Constances,  elle 
n'est  que  secondaire:  elle  doit  souvent  ceder  le  pas 
a  la  silrete,  a  la  solidite,  a  la  duree  de  la  machine, 
aupeu  de  place  qu'ilfaut  lui  faire  occuper,  au  peu 
defrais  de  son  etablissement,  etc.  Savoir  appre- 
cier,  dans  chaque  cas,  a  leur  juste  valeur,  les  con- 
siderations de  convenance  et  d 'economic  quipeuvent 
se  presenter  ;  savoir  discerner  les  plus  importantes 
de  celles  qui  sont  seulement  accessoires,  les  balancer 
toutes  convenablement  entre  elles,  afin  de  parvenir, 
par  les  moyens  les  plus  faciles,  au  meilleur  resul- 
tat: tel  doit  etre  le  principal  talent  de  I'homme 
appele  a  diriger,  a  co-ordonner  entre  eux  les  travaux 
de  ses  semblables,  a  les  faire  concourir  vers  un  but 
utile  de  quelque  genre  qu'il  soit." 

Such  was  the  work  and  such  the  character  of 
this  wonderful  man.  Those  whose  desire  to  fol- 
low more  closely  and  to  witness  the  process  of  de- 


THE   WORK  OF  1SADI  CARNOT.  19 

velopment  of  the  work  of  which  this  initial  paper 
of  Carnot  was  the  introductory,  should  study  the 
contribution  of  Sir  William  Thomson  to  this  devel- 
opment, as  published  in  1849, — a  paper  which 
constitutes  that  physicist  the  virtual  discoverer  of 
Carnot  and  the  godfather  of  the  man  and  his 
thoughts.  This  paper  constitutes  the  final  chapter 
of  this  little  book. 

From  that  time  the  additional  progress  so  rap- 
idly made  in  the  new  science  was  as  inevitable 
as  the  development  of  a  gold-field,  once  the  pre- 
cious metal  has  been  found  in  paying  quantities 
in  the  hitherto  unvisited  canons  and  gorges  of 
a  distant  and  unexplored  mountain-range.  But 
great  as  is  the  work  since  done,  and  great  as  have 
been  the  discoveries  and  the  discoverers  of  later 
years,  none  claims  our  gratitude  and  compels  our 
respect  in  greater  degree  than  does  the  original 
discoverer — 

SADI  CARNOT. 


II. 

LIFE  OF  SADI  CARNOT. 
BY  M.  H.  CARNOT. 

As  the  life  of  Sadi  Carnot  was  not  marked  by 
any  notable  event,  his  biography  would  have  occu- 
pied only  a  few  lines  ;  but  a  scientific  work  by  him, 
after  remaining  long  in  obscurity,  brought  again 
to  light  many  years  after  his  death,  has  caused  his 
name  to  be  placed  among  those  of  great  inventors. 
In  regard  to  his  person,  his  mind,  his  character, 
nothing  whatever  has  been  known.  Since  there  re- 
mains a  witness  of  his  private  life — the  sole  witness, 
has  he  not  a  duty  to  fulfil  ?  Ought  he  not  to 
satisfy  the  natural  and  legitimate  interest  which 
attaches  to  any  man  whose  work  has  deserved  a 
portion  of  glory  ? 

Nicolas-Leonard-Sadi  Carnot  was  born  Junel, 
1796,  in  the  smaller  Luxembourg.  This  was  that 
part  of  the  palace  where  our  father  then  dwelt  as 
a  member  of  the  Directory.  Our  father  had  a 
predilection  for  the  name  of  Sadi,  which  recalled 
to  his  mind  ideas  of  wisdom  and  poetry.  His  first- 
born  had  borne  this  name,  and  despite  the  fate 

20 


LIFE  OF  SADI  CARNOT.  21 

of  this  poor  child,  who  lived  but  a  few  months, 
he  called  the  second  also  Sad  I,  in  memory  of  the 
celebrated  Persian  poet  and  moralist. 

Scarcely  a  year  had  passed  when  the  proscrip- 
tion, which  included  the  Director,  obliged  him  to 
give  up  his  life,  or  at  least  his  liberty,  to  the  con- 
spirators of  fructidor.  Our  mother  carried  her 
son  far  from  the  palace  in  which  violation  of  law 
had  just  triumphed.  She  fled  to  St.  Omer,  with 
her  family,  while  her  husband  was  exiled  to  Switz- 
erland, then  to  Germany. 

Our  mother  often  said  to  me,  "  Thy  brother  was 
born  in  the  midst  of  the  cares  and  agitations  of 
grandeur,  thou  in  the  calm  of  an  obscure  retreat. 
Your  constitutions  show  this  difference  of  origin/' 

My  brother  in  fact  was  of  delicate  constitution. 
He  increased  his  strength  later,  by  means  of  va- 
ried and  judicious  bodily  exercises.  He  was  of 
medium  size,  endowed  with  extreme  sensibility 
and  at  the  same  time  with  extreme  energy,  more 
than  reserved,  almost  rude,  but  singularly  cou- 
rageous on  occasion.  When  he  felt  himself  to  be 
contending  against  injustice,  nothing  could  re- 
strain him.  The  following  is  an  anecdote  in  illus- 
tration. 

The  Directory  had  given  place  to  the  Consulate. 
Carnot,  after  two  years  of  exile,  returned  to  his 


22  LIFE  OF  SADI  CARNOT. 

country  and  was  appointed  Minister  of  War. 
Bonaparte  at  the  same  time  was  still  in  favor  with 
the  republicans.  He  remembered  that  Carnot  had 
assisted  him  in  the  beginning  of  his  military  ca- 
reer, and  he  resumed  the  intimate  relation  which 
had  existed  between  them  during  the  Directory. 
When  the  minister  went  to  Malmaison  to  work 
with  the  First  Consul,  he  often  took  with  him  his 
son,  then  about  four  years  old,  to  stay  with 
Madame  Bonaparte,  who  was  greatly  attached  to 
him. 

She  was  one  day  with  some  other  ladies  in  a 
small  boat  on  a  pond,  the  ladies  rowing  the  boat 
themselves,  when  Bonaparte,  unexpectedly  ap- 
pearing, amused  himself  by  picking  up  stones  and 
throwing  them  near  the  boat,  spattering  water  on 
the  fresh  toilets  of  the  rowers.  The  ladies  dared 
not  manifest  their  displeasure,  but  the  little  Sadi, 
after  having  looked  on  at  the  affair  for  some  time, 
suddenly  placed  himself  boldly  before  the  con- 
queror of  Marengo,  and  threatening  him  with  his 
fist,  he  cried  "Beast  of  a  First  Consul,  will  you 
stop  tormenting  those  ladies  I" 

Bonaparte,  at  this  unexpected  attack,  stopped 
and  looked  in  astonishment  at  the  child.  Then 
he  was  seized  with  a  fit  of  laughter  in  which  all 
the  spectators  of  the  scene  joined. 


LIFE  OF  SAD1  CARNOT.  23 

At  another  time,  when  the  minister,  wishing  to 
return  to  Paris,  sought  his  son,  who  had  been  left 
with  Madame  Bonaparte,  it  was  discovered  that  he 
had  run  away.  They  found  him  a  long  way  off,  in 
a  mill,  the  mechanism  of  which  he  was  trying  to 
understand.  This  desire  had  been  in  the  child's 
mind  for  days,  and  the  honest  miller,  not  knowing 
who  he  was,  was  kindly  answering  all  his  ques- 
tions. Curiosity,  especially  in  regard  to  mechanics 
and  physics,  was  one  of  the  essential  traits  of 
Sadies  mind. 

On  account  of  this  disposition  so  early  mani- 
fested, Carnot  did  not  hesitate  to  give  a  scientific 
direction  to  the  studies  of  his  son.  He  was  able 
to  undertake  this  task  himself  when  the  monarchi- 
cal tendencies  of  the  new  government  had  deter- 
mined him  to  retire.  For  a  few  months  only  Sadi 
followed  the  course  of  M.  Bourdon  at  the  Charle- 
magne Lycee  to  prepare  himself  for  the  Poly- 
technic School. 

The  pupil  made  rapid  progress.  He  was  just 
sixteen  years  old  when  he  was  admitted  to  the 
school,  the  twenty-fourth  on  the  list.  This  was 
in  1812.  The  following  year  he  left  it,  first  in 
artillery.  But  he  was  considered  too  young  for  the 
school  of  Metz,  and  he  continued  his  studies  at 
Paris  for  a  year.  To  this  circumstance  is  due  the 


24  LIFE  OF  SADI  CARNOT. 

fact  that  he  took  part  in  March,  1814,  in  the 
military  exploits  of  Vincennes,  and  not  of  the 
butte  Chaumont,  as  almost  all  the  historians  of 
the  siege  of  Paris  declared.  M.  Chasles,  one  of 
Sadi's  school-fellows,  took  pains  to  rectify  this 
error  at  a  seance  of  the  Institute  in  1869. 

If  the  pupils  of  the  Polytechnic  School  did  not 
earlier  enter  into  the  campaign,  it  was  not  because 
they  had  not  asked  to  do  so.  I  find  in  my  broth- 
er's papers  the  copy  of  an  address  to  the  Emperor, 
signed  by  them  December  29,  1813  : 

"  SIRE  :  The  country  needs  all  its  defenders. 
The  pupils  of  the  Polytechnic  School,  faithful  to 
their  motto,  ask  to  be  permitted  to  hasten  to  the 
frontiers  to  share  the  glory  of  the  brave  men  who 
are  consecrating  themselves  to  the  safety  of  France. 
The  battalion,  proud  of  having  contributed  to  the 
defeat  of  the  enemy,  will  return  to  the  school  to 
cultivate  the  sciences  and  prepare  for  new  services/' 

General  Carnot  was  at  Anvers,  which  he  had  just 
been  defending  against  the  confederate  English, 
Prussians,  and  Swedes,  where  the  French  flag  yet 
floated,  when  he  wrote  to  his  son,  .April  12,  1814  : 

"  MY  DEAR  SADI  :  I  have  learned  with  extreme 
pleasure  that  the  battalion  .of  the  Polytechnic 
School  has  distinguished  itself,  and  that  you  have 
performed  your  first  military  exploits  with  honor. 


LIFE  OF  SADI  CARNOT.  25 

When  I  am  recalled,  I  shall  be  very  glad  if  the 
Minister  of  War  will  give  you  permission  to  come 
to  me.  You  will  become  acquainted  with  a  fine 
country  and  a  beautiful  city,  where  I  have  had  the 
satisfaction  of  remaining  in  peace  while  disaster 
has  overwhelmed  so  many  other  places." 

Peace  being  restored,  Sadi  rejoined  his  father  at 
Anvers  and  returned  with  him  into  France. 

In  the  month  of  October  he  left  the  Polytech- 
nic School,  ranking  sixth  on  the  list  of  young 
men  destined  to  service  in  the  engineer  corps, 
and  went  to  Metz  as  a  cadet  sub-lieutenant  at  the 
school.  Many  scientific  papers  that  he  wrote  there 
were  a  decided  success.  One  is  particularly  re- 
ferred to  as  very  clever,  a  memoir  on  the  instru- 
ment called  the  theodolite  which  is  used  in  astron- 
omy and  geodesy. 

I  obtain  these  details  from  M.  Ollivier,  who  was 
of  the  same  rank  as  Sadi  and  who,  later,  was  one 
of  the  founders  of  the  EcoleCentrale.  Among  his 
other  comrades  besides  M.  Chasles,  the  learned 
geometrician  just  now  referred  to,  was  Gen.  Du- 
vivier,  lamented  victim  of  the  insurrection  of 
June  1848.  I  ought  also  to  mention  M  Robelin, 
Sadi's  most  intimate  friend,  who  came  to  help  me 
burse  him  during  his  last  illness,  and  who  pub- 


UNIVERSITY 


26  LIFE  OF  SADI  CABNOT. 

lished  a  notice  concerning  him  in  the  Revue  ency- 
clopedique,  t.  Iv. 

The  events  of  1815  brought  General  Carnot  back 
into  politics  during  the  "  Cent  Jours  "  which  ended 
in  a  fresh  catastrophe. 

This  gave  Sadi  a  glimpse  of  human  nature  of 
which  he  could  not  speak  without  disgust.  His 
little  sub-lieutenant's  room  was  visited  by  certain 
superior  officers  who  did  not  disdain  to  mount  to 
the  third  floor  to  pay  their  respects  to  the  son  of 
the  new  minister. 

Waterloo  put  an  end  to  their  attentions.  The 
Bourbons  re-established  on  the  throne,  Carnot  was 
proscribed  and  Sadi  sent  successively  into  many 
trying  places  to  pursue  his  vocation  of  engineer, 
to  count  bricks,  to  repair  walls,  and  to  draw  plans 
destined  to  be  hidden  in  portfolios.  He  performed 
these  duties  conscientiously  and  without  hope  of 
recompense,  for  his  name,  which  not  long  before 
had  brought  him  so  many  flatteries,  was  hence- 
forth the  cause  of  his  advancement  being  long 
delayed. 

In  1818  there  came  an  unlooked-for  royal  ordi- 
nance, authorizing  the  officers  of  all  branches  of 
the  service  to  present  themselves  at  the  examina- 
tions for  the  new  corps  of  the  staff.  Sadi  was 
well  aware  that  favor  had  much  more  to  do  with 


LIFE  OF  SADI  CARNOT.  27 

this  matter  than  ability,  but  he  was  weary  of 
garrison  life.  The  stay  in  small  fortresses  to 
which  the  nature  of  his  work  confined  him  did 
not  offer  sufficient  resources  to  his  love  of  study. 
Then  he  hoped,  and  his  hope  was  realized,  that  a 
request  for  a  furlough  would  be  obtained  without 
difficulty,  and  would  insure  him  the  leisure  that 
he  sought.  In  spite  of  the  friendly  opposition  of 
some  chiefs  of  the  engineer  corps,  testifying  to  a 
sincere  regret  at  the  removal  from  their  register 
of  a  name  which  had  gained  honor  among  them, 
Sadi  came  to  Paris  to  take  the  examination,  and 
was  appointed  lieutenant  on  the  staff,  January  20, 
1819. 

He  hastened  to  obtain  his  furlough,  and  availed 
himself  of  it  to  lead,  in  Paris  and  in  the  country 
round  about  Paris,  a  studious  life  interrupted  but 
once,  in  1821,  by  a  journey  to  Germany  to  visit  our 
father  in  his  exile  at  Magdeburg.  We  had  then 
the  pleasure  of  passing  some  weeks  all  three  to- 
gether. 

When,  two  years  later,  death  took  from  us  this 
revered  father  and  I  returned  alone  to  France,  I 
found  Sadi  devoting  himself  to  his  scientific  studies, 
which  he  alternated  with  the  culture  of  the  arts. 
In  this  way  also,  his  tastes  had  marked  out  for 
him  an  original  direction,  for  no  one  was  more 


28  LIFE  OF  SADI  CARNOT. 

opposed  than  he  to  the  traditional  and  the  con- 
ventional. On  his  music-desk  were  seen  only  the 
compositions  of  Lully  that  he  had  studied,  and 
the  concert!  of  Viotti  which  he  executed.  On  his 
table  were  seen  only  Pascal,  Moliere,  or  La  Fon- 
taine, and  he  knew  his  favorite  books  almost  by 
heart.  I  call  this  direction  original,  because  it 
was  anterior  to  the  artistic  and  literary  movement 
which  preceded  the  revolution  of  1830.  As  to  the 
sympathy  of  Sadi  for  the  author  of  the  Provin- 
ciates, it  was  due  not  only  to  the  respect  of  the 
young  mathematician  for  one  of  the  masters  of 
science,  but  his  devoutly  religious  mind  regarded 
with  horror  hypocrisy  and  hypocrites. 

Appreciating  the  useful  and  the  beautiful,  Sadi 
frequented  the  museum  of  the  Louvre  and  the 
Italian  Theatre,  as  well  as  the  Jardin  des  Plantes 
and  the  Conservatoire  des  Arts  et  Metiers.  Music 
was  almost  a  passion  with  him.  He  probably  in- 
herited this  from  our  mother,  who  was  an  excel- 
lent pianist,  to  whom  Dalayrac  and  especially 
Monsigny,  her  compatriot,  had  given  instruction. 
Not  content  with  being  able  to  play  well  on  the 
violin,  Sadi  carried  to  great  length  his  theoretical 
studies. 

His  insatiable  intellect,  moreover,  would  not 
allow  him  to  remain  a  stranger  to  any  branch  of 


LIFE  OF  8ADI  CARNOT.  29 

knowledge.  He  diligently  followed  the  course  of 
the  College  of  France  and  of  the  Sorbonne,  of 
the  Ecole  des  Mines,  of  the  Museum,  and  of  the 
Bibliotheque.  He  visited  the  workshops  with 
eager  interest,  and  made  himself  familiar  with  the 
processes  of  manufacture;  mathematical  sciences, 
natural  history,  industrial  art,  political  economy, 
— all  these  he  cultivated  with  equal  ardor.  I  have 
seen  him  not  only  practise  as  an  amusement,  but 
search  theoretically  into,  gymnastics,  fencing, 
swimming,  dancing,  and  even  skating.  In  even 
these  things  Sadi  acquired  a  superiority  which 
astonished  specialists  when  by  chance  he  forgot 
himself  enough  to  speak  of  them,  for  the  satisfac- 
tion of  his  own  mind  was  the  only  aim  that  he 
sought. 

He  had  such  a  repugnance  to  bringing  himself 
forward  that,  in  his  intimate  conversations  with  a 
few  friends,  he  kept  them  ignorant  of  the  treasures 
of  science  which  he  had  accumulated.  They  never 
knew  of  more  than  a  small  part  of  them.  How 
was  it  that  he  determined  to  formulate  his  ideas 
about  the  motive  power  of  heat,  and  especially  to 
publish  them  ?  I  still  ask  myself  this  question, — I, 
who  lived  with  him  in  the  little  apartment  where 
our  father  was  confined  in  the  Rue  du  Parc-Royal 
while  the  police  of  the  first  Restoration  were 


30  LIFE  OF  SADI  GARNOT. 

threatening  him.  Anxious  to  be  perfectly  clear, 
Sadi  made  me  read  some  passages  of  his  manu- 
script in  order  to  convince  himself  that  it  would 
be  understood  by  persons  occupied  with  other 
studies. 

Perhaps  a  solitary  life  in  small  garrisons,  in  the 
work-room  and  in  the  chemical  laboratory,  had 
increased  his  natural  reserve.  In  small  compa- 
nies, however,  he  was  not  at  all  taciturn.  He  took 
part  voluntarily  in  the  gayest  plays,  abandoning 
himself  to  lively  chat.  "The  time  passed  in 
laughing  is  well  spent,"  he  once  wrote.  His  lan- 
guage was  at  such  times  full  of  wit,  keen  without 
malice,  original  without  eccentricity,  sometimes 
paradoxical,  but  without  other  pretension  than 
that  of  an  innocent  activity  of  intelligence.  He 
had  a  very  warm  heart  under  a  cold  manner.  He 
was  obliging  and  devoted,  sincere  and  true  in  his 
dealings. 

Towards  the  end  of  1826,  a  new  royal  ordinance 
having  obliged  the  staff  lieutenants  to  return  to 
the  ranks,  Sadi  asked  and  obtained  a  return  to  the 
engineer  corps,  in  which  he  received  the  following 
year,  as  his  rank  of  seniority,  the  grade  of  captain. 

Military  service,  however,  weighed  upon  him. 
Jealous  of  his  liberty,  in  1828,  he  laid  aside  his 
uniform  that  he  might  be  free  to  come  and  go  at 


LIFE  OF  8ADI  CARNOT.  31 

will.  He  took  advantage  of  his  leisure  to  make 
journeys  and  to  visit  our  principal  centres  of 
industry. 

He  frequently  visited  M.  Clement  Desormes, 
professor  at  the  Conservatoire  des  Arts  et  Metiers, 
who  had  made  great  advances  in  applied  chemistry. 
M.  Desormes  willingly  took  counsel  with  him. 
He  was  a  native  of  Bourgogne,  our  family  coun- 
try, which  circumstance,  I  believe,  brought  them 
together. 

It  was  before  this  period  (in  1824)  that  Sadi  had 
published  his  Reflexions  sur  la  puissance  motrice, 
du  feu.  He  had  seen  how  little  progress  had  been 
made  in  the  theory  of  machines  in  which  this 
power  was  employed.  He  had  ascertained  that 
the  improvements  made  in  their  arrangement  were 
effected  tentatively,  and  almost  by  chance.  He 
comprehended  that  in  order  to  raise  this  important 
art  above  empiricism,  and  to  give  it  the  rank  of  a 
science,  it  was  necessary  to  study  the  phenomena 
of  the  production  of  motion  by  heat,  from  the 
most  general  point  of  view,  independently  of  any 
mechanism,  of  any  special  agent ;  and  such  had 
been  the  thought  of  his  life. 

Did  he  foresee  that  this  small  brochure  would 
become  the  foundation  of  a  new  science?  He 
tnust  have  attached  much  importance  to  it  to 


32  LIFE  OF  SADI  CARNOT. 

publish  it,  and  bring  himself  out  of  his  voluntary 
obscurity. 

In  fact  (as  his  working  notes  prove),  he  per- 
ceived the  existing  relation  between  heat  and 
mechanical  work ;  and  after  having  established  the 
principle  to  which  savants  have  given  his  name, 
he  devoted  himself  to  the  researches  which  should 
enable  him  to  establish  with  certainty  the  second 
principle,  that  of  equivalence,  which  he  already 
clearly  divined.  Thermodynamics  was  established 
from  that  time. 

But  these  researches  were  rudely  interrupted  by 
a  great  event — the  Revolution  of  July,  1830. 

Sadi  welcomed  it  enthusiastically — not,  however, 
it  is  evident,  as  a  personal  advantage. 

Several  old  members  of  the  Convention  were 
still  living,  even  of  those  who  had  become  cele- 
brated ;  no  favor  of  the  new  government  was 
accorded  them.  To  the  son  of  Philippe-Egalite 
was  ascribed  a  saying  which,  if  it  was  untrue,  at 
least  agreed  well  with  the  sentiment  of  his  posi- 
tion: "I  can  do  nothing  for  the  members  of  the 
Convention  themselves,"  he  said,  "but  for  their 
families  whatever  they  will." 

However  it  may  be,  some  of  those  about  him 
vaguely  questioned  my  brother  as  to  his  desires  in 
case  one  of  us  should  be  called  to  the  Chamber  of 


LIFE  OF  SADI  CARNOT.  33 

Peers,  of  which  Carnot  had  been  a  member  in 
1815.  We  had  on  this  occasion  a  brief  conference. 
Unknown  to  us  both,  this  distinction  could  be 
offered  only  to  a  title  in  some  sort  hereditary. 
We  could  not  accept  it  without  forsaking  the  prin- 
ciples of  Carnot,  who  had  combated  the  heredity 
of  the  peerage.  The  paternal  opinion  therefore 
came  to  second  our  distaste  for  the  proposition, 
and  dictated  our  reply. 

Sadi  frequented  the  popular  reunions  at  this 
period  without  forsaking  his  role  of  a  simple  ob- 
server. 

Nevertheless  he  was,  when  occasion  demanded 
it,  a  man  of  prompt  and  energetic  action.  One 
incident  will  suffice  to  prove  this,  and  to  show  the 
sang-froid  which  characterized  him. 

On  the  day  of  the  funeral  of  Gen.  Lamarque, 
Sadi  was  walking  thoughtfully  in  the  vicinity  of 
the  insurrection.  A  horseman  preceding  a  com- 
pany, and  who  was  evidently  intoxicated,  passed 
along  the  street  on  the  gallop,  brandishing  his 
sabre  and  striking  down  the  passers-by.  Sadi 
darted  forward,  cleverly  avoided  the  weapon  of 
the  soldier,  seized  him  by  the  leg,  threw  him  to 
the  earth  and  laid  him  in  the  gutter,  then  contin- 
ued on  his  way  to  escape  from  the  cheers  of  the 
crowd,  amazed  at  this  daring  deed. 


34  LIFE  OF  SADI  CARNOT. 

Before  1830,  Sadi  had  formed  part  of  a  Reunion 
poly  technique  industrielle,  made  up  of  old  pupils 
of  the  school,  with  a  plan  of  study  in  common. 
After  1830,  he  was  a  member  of  the  Association 
polyteclmique,  consisting  also  of  graduates,  the 
object  being  the  popular  propagation  of  useful 
knowledge.  The  president  of  this  association  was 
M.  de  Choiseul-Praslin;  the  vice-presidents,  MM. 
de  Tracy,  Auguste  Comte,  etc. 

The  hopes  of  the  democracy  meanwhile  seeming 
to  be  in  abeyance,  Sadi  devoted  himself  anew  to 
study,  and  pursued  his  scientific  labors  with  all  the 
greater  energy,  as  he  brought  to  bear  upon  them 
the  political  ardor  now  so  completely  repressed. 
He  undertook  profound  researches  on  the  physical 
properties  of  gases  and  vapors,  and  especially  on 
their  elastic  tensions.  Unfortunately,  the  tables 
which  he  prepared  from  his  comparative  experi- 
ments were  not  completed;  but  happily  the  excel- 
lent works  of  Victor  Regnault,  so  remarkable  for 
their  accuracy,  have  supplied  to  science,  in  this 
respect,  the  blanks  of  which  Sadi  Carnot  was  con- 
scious. 

His  excessive  application  affected  his  health 
towards  the  end  of  June,  1832.  Feeling  temporar- 
ily better,  he  wrote  gayly  to  one  of  his  friends  who 
had  written  several  letters  to  him :  "My  delay  this 


LIFE  OF  SADI  CARNOT.  35 

time  is  not  without  excuse.  I  have  been  sick  for 
a  long  time,  and  in  a  very  wearisome  way.  I  have 
had  an  inflammation  of  the  lungs,  followed  by  scar- 
let-fever. (Perhaps  you  know  what  this  horrible 
disease  is.)  I  had  to  remain  twelve  days  in  bed, 
without  sleep  or  food,  without  any  occupation, 
amusing  myself  with  leeches,  with  drinks,  with 
baths,  and  other  toys  out  of  the  same  shop.  This 
little  diversion  is  not  yet  ended,  for  I  am  still  very 
feeble." 

This  letter  was  written  at  the  end  of  July. 

There  was  a  relapse,  then  brain  fever;  then  final- 
ly, hardly  recovered  from  so  many  violent  illnesses 
which  had  Aveakened  him  morally  and  physically, 
Sadi  was  carried  off  in  a  few  hours,  August  24, 
1832,  by  an  attack  of  cholera.  Towards  the  last, 
and  as  if  from  a  dark  presentiment,  he  had  given 
much  attention  to  the  prevailing  epidemic,  follow- 
ing its  course  with  the  attention  and  penetration 
that  he  gave  to  everything. 

Sadi  Carnot  died  in  the  vigor  of  life,  in  the 
brightness  of  a  career  that  he  bade  fair  to  run  with 
glory,  leaving  memory  of  profound  esteem  and 
affection  in  the  hearts  of  many  friends. 

His  copy-books,  filled  with  memoranda,  attest 
the  activity  of  his  mind,  the  variety  of  his  knowl- 
edge, his  love  of  humanity,  his' clear  sentiments  of 


36  LIFE  OF  SADI  CARNOT. 

justice  and  of  liberty.  We  can  follow  therein  the 
traces  of  all  his  various  studies.  But  the  only 
work  that  he  actually  completed  is  this  which  is 
here  published.  It  will  suffice  to  preserve  his 
name  from  oblivion. 

His  moral  character  has  other  claims  on  our 
recognition.  Our  only  ambition  here  is  to  present 
a  sketch  of  it.  But,  much  better  than  through 
the  perusal  of  these  few  pages,  Sadi  Carnot  can  be 
appreciated  by  reading  the  thoughts  scattered 
through  his  memoranda,  which  are  to  be  carefully 
collected.  There  are  many  practical  rules  of  con- 
duct which  he  records  for  himself  ;  many  observa- 
tions that  he  desires  to  fix  in  his  memory ;  some- 
times an  impression  that  has  just  come  to  him, 
grave  or  gay  ;  sometimes  too,  though  rarely,  a 
trace  of  ill-humor  directed  against  men  or  society. 
He  never  thought  that  these  notes,  the  outpouring 
of  his  mind,  would  be  read  by  other  eyes  than  his 
own,  or  that  they  would  some  day  be  used  to  judge 
him.  I  find  in  them,  for  my  part,  touching  analo- 
gies with  the  thoughts  of  my  father,  although  the 
father  and  son  had,  unfortunately,  lived  almost 
always  apart,  by  force  of  circumstances.* 

*  See  the  Appendix  for  these  memoranda,  and  for  other 
previously  unpublished  matter. 


III. 

REFLECTIONS  ON  THE  MOTIVE-POWER  OF 
HEAT,  AND  ON  MACHINES  FITTED  TO 
DEVELOP  THAT  POWER.* 

BY  S.  CARNOT. 

EVERY  one  knows  that  heat  can  produce  motion. 
That  it  x  possesses  vast  motive-power  no  one  can 
doubt,  in  these  days  when  the  steam-engine  is 
everywhere  so  well  known. 

To  heat  also  are  due  the  vast  movements  which 
take  place  on  the  earth.  It  causes  the  agitations 
of  the  atmosphere,  the  ascension  of  clouds,  the  fall 
of  rain  and  of  meteors,  the  currents  of  water  which 
channel  the  surface  of  the  globe,  and  of  which 

*  Sadi  Carnot's  Reflexions  sur  la  puissance  motrice  du 
feu  (Paris,  Bachelier  1824)  was  long  ago  completely  ex- 
hausted. As  but  a  small  number  of  copies  were  printed, 
this  remarkable  work  remained  long  unknown  to  the 
earlier  writers  on  Thermodynamics.  It  was  therefore  for 
the  benefit  of  savants  unable  to  study  a  work  out  of  print, 
as  well  as  to  render  honor  to  the  memory  of  Sadi  Carnot, 
that  the  new  publishers  of  the  Annales  Scientifique  de 
VEcole  Normale  superieure  (ii.  series,  1. 1,  1872)  published  a 
new  edition,  from  which  this  translation  is  reproduced. 

37 


38  MOTIVE  POWER  OF  HEAT. 

man  has  thus  far  employed  but  a  small  portion. 
Even  earthquakes  and  volcanic  eruptions  are  the 
result  of  heat. 

From  this  immense  reservoir  we  may  draw  the 
moving  force  necessary  for  our  purposes.  Nature, 
in  providing  us  with  combustibles  on  all  sides, 
has  given  us  the  power  to  produce,  at  all  times  and 
in  all  places,  heat  and  the  impelling  power  which 
is  the  result  of  it.  To  develop  this  power,  to 
appropriate  it  to  our  uses,  is  the  object  of  heat- 
engines. 

The  study  of  these  engines  is  of  the  greatest 
interest,  their  importance  is  enormous,  their  use 
is  continually  increasing,  and  they  seem  destined 
to  produce  a  great  revolution  in  the  civilized  world. 

Already  the  steam-engine  works  our  mines,  im- 
pels our  ships,  excavates  our  ports  and  our  rivers, 
forges  iron,  fashions  wood,  grinds  grains,  spins 
and  weaves  our  cloths,  transports  the  heaviest 
burdens,  etc.  It  appears  that  it  must  some  day 
serve  as  a  universal  motor,  and  be  substituted  for 
animal  power,  waterfalls,  and  air  currents. 

Over  the  first  of  these  motors  it  has  the  advan- 
tage of  economy,  over  the  two  others  the  inestima- 
ble advantage  that  it  can  be  used  at  all  times  and 
places  without  interruption. 

If,  some  day,  the  steam-engine  shall  be  so  per- 


MOTIVE  POWER  OF  HEAT.  39 

fected  that  it  can  be  set  up  and  supplied  with  fuel 
at  small  cost,  it  will  combine  all  desirable  qualities, 
and  will  afford  to  the  industrial  arts  a  range  the 
extent  of  which  can  scarcely  be  predicted.  It  is 
not  merely  that  a  powerful  and  convenient  motor 
that  can  be  procured  and  carried  anywhere  is 
substituted  for  the  motors  already  in  use,  but  that 
it  causes  rapid  extension  in  the  arts  in  which  it  is 
applied,  and  can  even  create  entirely  new  arts. 

The  most  signal  service  that  the  steam-engine 
has  rendered  to  England  is  undoubtedly  the 
revival  of  the  working  of  the  coal-mines,  which  had 
declined,  and  threatened  to  cease  entirely,  in  con- 
sequence of  the  continually  increasing  difficulty  of 
drainage,  and  of  raising  the  coal.*  We  should 
rank  second  the  benefit  to  iron  manufacture,  both 
by  the  abundant  supply  of  coal  substituted  for 
wood  just  when  the  latter  had  begun  to  grow  scarce, 

*It  may  be  said  that  coal-mining  has  increased  tenfold 
in  England  since  the  invention  of  the  steam-engine.  It  is 
almost  equally  true  in  regard  to  the  mining  of  copper,  tin, 
and  iron.  The  results  produced  in  a  half-century  by  the 
steam-engine  in  the  mines  of  England  are  to-day  parallel- 
ed in  the  gold  and  silver  mines  of  the  New  World — mines 
of  which  the  working  declined  from  day  to  day,  prin- 
cipally on  account  of  the  insufficiency  of  the  motors  em- 
ployed in  the  draining  and  the  extraction  of  the  minerals. 


40  MOTIVE  POWER  OF  HEAT. 

and  by  the  powerful  machines  of  all  kinds,  the  use 
of  which  the  introduction  of  the  steam-engine  has 
permitted  or  facilitated. 

Iron  and  heat  are,  as  we  know,  the  supporters, 
the  bases,  of  the  mechanic  arts.  It  is  doubtful  if 
there  be  in  England  a  single  industrial  establish- 
ment of  which  the  existence  does  not  depend  on 
the  use  of  these  agents,  and  which  does  not  freely 
employ  them.  To  take  away  to-day  from  England 
her  steam-engines  would  be  to  take  away  at  the 
same  time  her  coal  and  iron.  It  would  be  to  dry 
up  all  her  sources  of  wealth,  to  ruin  all  on  which 
her  prosperity  depends,  in  short, 'to  annihilate  that 
colossal  power.  The  destruction  of  her  navy, 
which  she  considers  her  strongest  defence,  would 
perhaps  be  less  fatal. 

The  safe  and  rapid  navigation  by  steamships 
may  be  regarded  as  an  entirely  new  art  due  to  the 
steam-engine.  Already  this  art  has  permitted  the 
establishment  of  prompt  and  regular  communica- 
tions across  the  arms  of  the  sea,  and  on  the  great 
rivers  of  the  old  and  new  continents.  It  has  made 
it  possible  to  traverse  savage  regions  where  before 
we  could  scarcely  penetrate.  It  has  enabled  us  to 
carry  the  fruits  of  civilization  over  portions  of  the 
globe  where  they  would  else  have  been  wanting  for 
years.  Steam  navigation  brings  nearer  together 


MOTIVE  POWER  OF  HEAT.  41 

the  most  distant  nations.  It  tends  to  unite  the 
nations  of  the  earth  as  inhahitants  of  one  country. 
In  fact,  to  lessen  the  time,  the  fatigues,  the  uncer- 
tainties, and  the  dangers  of  travel — is  not  this  the 
same  as  greatly  to  shorten  distances?* 

The  discovery  of  the  steam-engine  owed  its  birth, 
like  most  human  inventions,  to  rude  attempts 
which  have  been  attributed  to  different  persons, 
while  the  real  author  is  not  certainly  known.  It 
is,  however,  less  in  the  first  attempts  that  the  prin- 
cipal discovery  consists,  than  in  the  successive  im- 
provements which  have  brought  steam-engines  to 
the  condition  in  which  we  find  them  to-day.  There 
is  almost  as  great  a  distance  between  the  first  appa- 
ratus in  which  the  expansive  force  of  steam  was 
displayed  and  the  existing  machine,  as  between  the 
first  raft  that  man  ever  made  and  the  modern  vessel. 

If  the  honor  of  a  discovery  belongs  to  the  nation 
in  which  it  has  acquired  its  growth  and  all  its 
developments,  this  honor  cannot  be  here  refused 

*  We  say,  to  lessen  the  dangers  of  journeys.  In  fact, 
although  the  use  of  the  steam-engine  on  ships  is  attended 
by  some  danger  which  has  been  greatly  exaggerated,  this 
is  more  than  compensated  by  the  power  of  following  al- 
ways an  appointed  and  well-known  route,  of  resisting  the 
force  of  the  winds  which  would  drive  the  ship  towards 
the  shore,  the  shoals,  or  the  rocks. 


42  MOTIVE  POWER  OF  HEAT. 

to  England.  Savery,  Newcomen,  Smeaton,  the 
famous  Watt,  Woolf,  Trevithick,  and  some  other 
English  engineers,  are  the  veritable  creators  of  the 
steam-engine.  It  has  acquired  at  their  hands  all 
its  successive  degrees  of  improvement.  Finally,  it 
is  natural  that  an  invention  should  have  its  birth 
and  especially  be  developed,  be  perfected,  in  that 
place  where  its  want  is  most  strongly  felt. 

Notwithstanding  the  work  of  all  kinds  done  by 
steam-engines,  notwithstanding  the  satisfactory 
condition  to  which  they  have  been  brought  to-day, 
their  theory  is  very  little  understood,  and  the  at- 
tempts to  improve  them  are  still  directed  almost 
by  chance. 

The  question  has  often  been  raised  whether  the 
motive  power  of  heat*  is  unbounded,  whether  the 
possible  improvements  in  steam-engines  have  an 
assignable  limit, — a  limit  which  the  nature  of 
things  will  not  allow  to  be  passed  by  any  means 
whatever ;  or  whether,  on  the  contrary,  these  im- 
provements may  be  carried  on  indefinitely.  We 

*  We  use  here  the  expression  motive  power  to  express 
the  useful  effect  that  a  motor  is  capable  of  producing. 
This  effect  can  always  be  likened  to  the  elevation  of  a 
weight  to  a  certain  height.  It  has,  as  we  know,  as  a 
measure,  the  product  of  the  weight  multiplied  by  the 
height  to  which  it  is  raised. 


MOTIVE  POWER  OF  HEAT.  43 

have  long  sought,  and  are  seeking  to-day,  to  ascer- 
tain whether  there  are  in  existence  agents  preferable 
to  the  vapor  of  water  for  developing  the  motive 
power  of  heat;  whether  atmospheric  air,  for  ex- 
ample, would  not  'present  in  this  respect  great  ad- 
vantages. We  propose  now  to  submit  these  ques- 
tions to  a  deliberate  examination. 

The  phenomenon  of  the  production  of  motion 
by  heat  has  not  been  considered  from  a  sufficiently 
general  point  of  view.  We  have  considered  it  only 
in  machines  the  nature  and  mode  of  action  of 
which  have  not  allowed  us  to  take  in  the  whole 
extent  of  application  of  which  it  is  susceptible. 
In  such  machines  the  phenomenon  is,  in  a  way, 
incomplete.  It  becomes  difficult  to  recognize  its 
principles  and  study  its  laws. 

In  order  to  consider  in  the  most  general  way 
the  principle  of  the  production  of  motion  by  heat, 
it  must  be  considered  independently  of  any  mecha- 
nism or  any  particular  agent.  It  is  necessary  to 
establish  principles  applicable  not  only  to  steam- 
engines*  but  to  all  imaginable  heat-engines,  what- 


*  We  distinguish  here  the  steam-engine  from  the  heat- 
engine  in  general.  The  latter  may  make  use  of  any  agent 
whatever,  of  the  vapor  of  water  or  of  any  other,  to  develop 
the  motive  power  of  heat, 


44  MOTIVE  POWER  OF  HEAT. 

ever  the  working  substance  and  whatever  the 
method  by  which  it  is  operated. 

Machines  which  do  not  receive  their  motion  from 
heat,  those  which  have  for  a  motor  the  force  of 
men  or  of  animals,  a  waterfall,  an  air-current,  etc., 
can  be  studied  even  to  their  smallest  details  by 
the  mechanical  theory.  All  cases  are  foreseen,  all 
imaginable  movements  are  referred  to  these  general 
principles,  firmly  established,  and  applicable  under 
all  circumstances.  This  is  the  character  of  a  com- 
plete theory.  A  similar  theory  is  evidently  needed 
for  heat-engines.  We  shall  have  it  only  when  the 
laws  of  Physics  shall  be  extended  enough,  general- 
ized enough,  to  make  known  beforehand  all  the 
effects  of  heat  acting  in  a  determined  manner  on 
any  body. 

We  will  suppose  in  what  follows  at  least  a 
superficial  knowledge  of  the  different  parts  which 
compose  an  ordinary  steam-engine;  and  we  con- 
sider it  unnecessary  to  explain  what  are  the 
furnace,  boiler,  steam-cylinder,  piston,  condenser, 
etc. 

The  production  of  motion  in  steam-engines  is 
always  accompanied  by  a  circumstance  on  which 
we  should  fix  our  attention.  This  circumstance 
is  the  re-establishing  of  equilibrium  in  the  caloric; 
that  is,  its  passage  from  a  body  in  which  the 


MOTIVE  POWER  OF  HEAT.  45 

temperature  is  more  or  less  elevated,  to  another  in 
which  it  is  lower.  What  happens  in  fact  in  a 
steam-engine  actually  in  motion?  The  caloric 
developed  in  the  furnace  by  the  effect  of  the  com- 
bustion traverses  the  walls  of  the  boiler,  produces 
steam,  and  in  some  way  incorporates  itself  with  it. 
The  latter  carrying  it  away,  takes  it  first  into  the 
cylinder,  where  it  performs  some  function,  and 
from  thence  into  the  condenser,  where  it  is  lique- 
fied by  contact  with  the  cold  water  which  it  en- 
counters there.  Then,  as  a  final  result,  the  cold 
water  of  the  condenser  takes  possession  of  the 
caloric  developed  by  the  combustion.  It  is  heated 
by  the  intervention  of  the  steam  as  if  it  had  been 
placed  directly  over  the  furnace.  The  steam  is 
here  only  a  means  of  transporting  the  caloric. 
It  fills  the  same  office  as  in  the  heating  of  baths 
by  steam,  except  that  in  this  case  its  motion  is 
rendered  useful. 

We  easily  recognize  in  the  operations  that  we 
have  just  described  the  re-establishment  of  equi- 
librium in  the  caloric,  its  passage  from  a  more  or 
less  heated  body  to  a  cooler  one.  The  first  of 
these  bodies,  in  this  case,  is  the  heated  air  of  the 
furnace;  the  second  is  the  condensing  water.  The 
re-establishment  of  equilibrium  of  the  caloric 
takes  place  between  them,  if  not  completely,  at 


46  MOTIVE  POWER  OF  HEAT. 

least  partially,  for  on  the  one  hand  the  heated  air, 
after  having  performed  its  function,  having  passed 
round  the  boiler,  goes  out  through  the  chimney 
with  a  temperature  much  below  that  which  it  had 
acquired  as  the  effect  of  combustion;  and  on  the 
other  hand,  the  water  of  the  condenser,  after  hav- 
ing liquefied  the  steam,  leaves  the  machine  with 
a  temperature  higher  than  that  with  which  it 
entered. 

The  production  of  motive  power  is  then  due  in 
steam-engines  not  to  an  actual  consumption  of 
caloric,  but  to  its  transportation  from  a  warm 
body  to  a  cold  body,  that  is,  to  its  re-establishment 
of  equilibrium — an  equilibrium  considered  as  de- 
stroyed by  any  cause  whatever,  by  chemical  action 
such  as  combustion,  or  by  any  other.  We  shall 
see  shortly  that  this  principle  is  applicable  to 
any  machine  set  in  motion  by  heat. 

According  to  this  principle,  the  production  of 
heat  alone  is  not  sufficient  to  give  birth  to  the 
impelling  power:  it  is  necessary  that  there  should 
also  be  cold;  without  it,  the  heat  would  be  use- 
less. And  in  fact,  if  we  should  find  about  us 
only  bodies  as  hot  as  our  furnaces,  how  can  we 
condense  steam  ?  What  should  we  do  with  it  if 
once  produced  ?  We  should  not  presume  that  we 
might  discharge  it  into  the  atmosphere,  as  is  done 


MOTIVE  POWER  OF  HEAT.  47 

in  some  engines;*  the  atmosphere  would  not  re- 
ceive it.  It  does  receive  it  under  the  actual  con- 
dition of  things,  only  because  it  fulfils  the  office 
of  a  vast  condenser,  because  it  is  at  a  lower  tem- 
perature; otherwise  it  would  soon  become  fully 
charged,  or  rather  would  be  already  saturated,  f 

*  Certain  engines  at  high  pressure  throw  the  steam  out 
iuto  the  atmosphere  instead  of  the  condenser.  They  are 
used  specially  in  places  where  it  would  be  difficult  to 
procure  a  stream  of  cold  water  sufficient  to  produce 
condensation. 

f  The  existence  of  water  in  the  liquid  state  here 
necessarily  assumed,  since  without  it  the  steam-engine 
could  not  be  fed,  supposes  the  existence  of  a  pressure 
capable  of  preventing  this  water  from  vaporizing,  con- 
sequently of  a  pressure  equal  or  superior  to  the  tension 
of  vapor  at  that  temperature.  If  such  a  pressure  were 
not  exerted  by  the  atmospheric  air,  there  would  be  in- 
stantly produced  a  quantity  of  steam  sufficient  to  give 
rise  to  that  tension,  and  it  would  be  necessary  always 
to  overcome  this  pressure  iu  order  to  throw  out  the 
steam  from  the  engines  into  the  new  atmosphere.  Now 
this  is  evidently  equivalent  to  overcoming  the  tension 
which  the  steam  retains  after  its  condensation,  as  effected 
by  ordinary  means. 

If  a  very  high  temperature  existed  at  the  surface  of 
our  globe,  as  it  seems  certain  that  it  exists  in  its  interior, 
all  the  waters  of  the  ocean  would  be  in  a  state  of  vapor 
in  the  atmosphere,  and  no  portion  of  it  would  be  found 
in  a  liquid  state. 


48  MOTIVE  POWER  OF  HEAT. 

Wherever  there  exists  a  difference  of  tempera- 
ture, wherever  it  has  been  possible  for  the  equilib- 
rium of  the  caloric  to  be  re-established,,  it  is  possible 
to  have  also  the  production  of  impelling  power. 
Steam  is  a  means  of  realizing  this  power,  but  it  is 
not  the  only  one.  All  substances  in  nature  can 
be  employed  for  this  purpose,,  all  are  susceptible  of 
changes  of  volume,  of  successive  contractions  and 
dilatations,  through  the  alternation  of  heat  and  cold. 
All  are  capable  of  overcoming  in  their  changes  of 
volume  certain  resistances,  and  of  thus  developing 
the  impelling  power.  A  solid  body — a  metallic 
bar  for  example — alternately  heated  and  cooled  in- 
creases and  diminishes  in  length,  and  can  move 
bodies  fastened  to  its  ends.  A  liquid  alternately 
heated  and  cooled  increases  and  diminishes  in  vol- 
ume, and  can  overcome  obstacles  of  greater  or  less 
size,  opposed  to  its  dilatation.  An  aeriform  fluid  is 
susceptible  of  considerable  change  of  volume  by 
variations  of  temperature.  If  it  is  enclosed  in  an 
expansible  space,  such  as  a  cylinder  provided  with 
a  piston,  it  will  produce  movements  of  great  ex- 
tent. Vapors  of  all  substances  capable  of  passing 
into  a  gaseous  condition,  as  of  alcohol,  of  mercury, 
of  sulphur,  etc.,  may  fulfil  the  same  office  as  vapor 
of  water.  The  latter,  alternately  heated  and 
cooled,  would  produce  motive  power  in  the  shape 


UNIVERSITY 


MOTIVE  POWER  OF  HEAT.  49 

of  permanent  gases,  that  is,  without  ever  return- 
ing to  a  liquid  state.  Most  of  these  substances 
have  been  proposed,  many  even  have  been  tried, 
although  up  to  this  time  perhaps  without  remark- 
able success. 

We  have  shown  that  in  steam-engines  the  motive- 
power  is  due  to  a  re-  establishment  of  equilibrium 
in  the  caloric  ;  this  takes  place  not  only  for  steam- 
engines,  but  also  for  every  heat-engine  —  that  is, 
for  every  machine  of  which  caloric  is  the  motor. 
Heat  can  evidently  be  a  cause  of  motion  only  by 
virtue  of  the  changes  of  volume  or  of  form  which 
it  produces  in  bodies. 

These  changes  are  not  caused  by  uniform  tem- 
perature, but  rather  by  alternations  of  heat  and 
cold.  Now  to  heat  any  substance  whatever  requires 
a  body  warmer  than  the  one  to  be  heated;  to  cool 
it  requires  a  cooler  body.  We  supply  caloric  to 
the  first  of  these  bodies  that  we  may  transmit 
it  to  the  second  by  means  of  the  intermediary 
substance.  This  is  to  re-establish,  or  at  least  to 
endeavor  to  re-establish,  the  equilibrium  of  the 
caloric. 

It  is  natural  to  ask  here  this  curious  and  impor- 
tant question  :  Is  the  motive  power  of  heat  invari- 
able in  quantity,  or  does  it  vary  with  the  agent 
employed  to  realize  it  as  the  intermediary  sub- 


50  MOTIVE  POWER  OF  HEAT. 

stance,  selected  as  the  subject  of  action  of   the 
heat? 

It  is  clear  that  this  question  can  be  asked  only 
in  regard  to  a  given  quantity  of  caloric,*  the  differ- 
ence of  the  temperatures  also  being  given.  We 
take,  for  example,  one  body  A  kept  at  a  tempera- 
ture of  100°  and  another  body  B  kept  at  a  tempera- 
ture of  0°,  and  ask  what  quantity  of  motive  power 
can  be  produced  by  the  passage  of  a  given  portion 
of  caloric  (for  example,  as  much  as  is  necessary  to 
melt  a  kilogram  of  ice)  from  the  first  of  these 
bodies  to  the  second.  We  inquire  whether  this 
quantity  of  motive  power  is  necessarily  limited, 
whether  it  varies  with  the  substance  employed  to 
realize  it,  whether  the  vapor  of  water  offers  in  this 
respect  more  or  less  advantage  than  the  vapor  of 
alcohol,  of  mercury,  a  permanent  gas,  or  any  other 
substance.  We  will  try  to  answer  these  questions, 
availing  ourselves  of  ideas  already  established. 

*  It  is  considered  unnecessary  to  explain  here  what  is 
quantity  of  caloric  or  quantity  of  heat  (for  we  employ 
these  two  expressions  indifferently),  or  to  describe  how  we 
measure  these  quantities  by  the  calorimeter.  Nor  will  we 
explain  what  is  meant  by  latent  heat,  degree  of  temperature, 
specific  heat,  etc.  The  reader  should  be  familiarized  with 
these  terms  through  the  study  of  the  elementary  treatises 
of  physics  or  of  chemistry. 


MOTIVE  POWER  OF  HEAT.  51 

We  have  already  remarked  upon  this  self-evident 
factj  or  fact  which  at  least  appears  evident  as  soon 
as  we  reflect  on  the  changes  of  volume  occasioned 
by  heat  :  wherever  there  exists  a  difference  of  tem- 
perature, motive-power  can  be  produced.  Recipro- 
cally, wherever  we  can  consume  this  power,  it  is 
possible  to  produce  a  difference  of  temperature, 
it  is  possible  to  occasion  destruction  of  equilibrium 
in  the  caloric.  Are  not  percussion  and  the  fric- 
tion of  bodies  actually  means  of  raising  their  tem- 
perature, of  making  it  reach  spontaneously  a 
higher  degree  than  that  of  the  surrounding  bodies, 
and  consequently  of  producing  a  destruction  of 
equilibrium  in  the  caloric,  where  equilibrium  pre- 
viously existed  ?  It  is  a  fact  proved  by  experience, 
that  the  temperature  of  gaseous  fluids  is  raised  by 
compression  and  lowered  by  rarefaction.  This  is 
a  sure  method  of  changing  the  temperature  of 
bodies,  and  destroying  the  equilibrium  of  the 
caloric  as  many  times  as  may  be  desired  with  the 
same  substance.  The  vapor  of  water  employed  in 
an  inverse  manner  to  that  in  which  it  is  used  in 
steam-engines  can  also  be  regarded  as  a  means  of 
destroying  the  equilibrium  of  the  caloric.  To  be 
convinced  of  this  we  need  but  to  observe  closely 
the  manner  in  which  motive  power  is  developed  by 
the  action  of  heat  on  vapor  of  water.  Imagine 


52  MOTIVE  POWER  OF  HEAT. 

two  bodies  A  and  B,  kept  each  at  a  constant  tem- 
perature, that  of  A  being  higher  than  that  of  B. 
These  two  bodies,  to  which  we  can  give  or  from 
which  we  can  remove  the  heat  without  causing 
their  temperatures  to  vary,  exercise  the  functions 
of  two  unlimited  reservoirs  of  caloric.  We  will 
call  the  first  the  furnace  and  the  second  the  re- 
frigerator. 

If  we  wish  to  produce  motive  power  by  carrying 
a  certain  quantity  of  heat  from  the  body  A  to  the 
body  B  we  shall  proceed  as  follows  : 

(1)  To  borrow  caloric  from  the  body  A  to  make 
steam   with  it — that  is,  to  make  this  body  fulfil 
the  function  of  a  furnace,  or  rather  of  the  metal 
composing  the  boiler  in  ordinary  engines — we  here 
assume  that  the  steam  is  produced  at  the  same 
temperature  as  the  body  A. 

(2)  The  steam  having  been  received  in  a  space 
capable  of  expansion,  such  as  a  cylinder  furnished 
with  a  piston,  to  increase  the  volume  of  this  space, 
and  consequently  also  that  of  the  steam.  Thus  rare- 
fied, the  temperature   will  fall  spontaneously,  as 
occurs  with  all  elastic  fluids  ;  admit  that  the  rare- 
faction may  be  continued  to  the  point  where  the 
temperature  becomes  precisely  that  of  the  body  B. 

(3)  To  condense  the  steam  by  putting  it  in  con- 
tact with  the  body  B,  and  at  the  same  time  exert- 


MOTIVE  POWER  OF  HEAT.  53 

ing  on  it  a  constant  pressure  until  it  is  entirely 
liquefied.  The  body  B  fills  here  the  place  of  the 
injection-water  in  ordinary  engines,  with  this  dif- 
ference, that  it  condenses  the  vapor  without 
mingling  with  it,  and  without  changing  its  own 
temperature.* 

*  We  may  perhaps  wonder  here  that  the  body  B  being 
at  the  same  temperature  as  the  steam  is  able  to  condense 
it.  Doubtless  this  is  not  strictly  possible,  but  the  slightest 
difference  of  temperature  will  determine  the  condensation, 
which  suffices  to  establish  the  justice  of  our  reasoning.  It 
is  thus  that,  in  the  differential  calculus,  it  is  sufficient  that 
we  can  conceive  the  neglected  quantities  indefinitely  re- 
ducible in  proportion  to  the  quantities  retained  in  the 
equations,  to  make  certain  of  the  exact  result. 

The  body  B  condenses  the  steam  without  changing  its 
own  temperature — this  results  from  our  supposition.  We 
have  admitted  that  this  body  may  be  maintained  at  a  con- 
stant temperature.  We  take  away  the  caloric  as  the  steam 
furnishes  it.  This  is  the  condition  in  which  the  metal  of 
the  condenser  is  found  when  the  liquefaction  of  the  steam 
is  accomplished  by  applying  cold  water  externally,  as  was 
formerly  done  in  several  engines.  Similarly,  the  water  of 
a  reservoir  can  be  maintained  at  a  constant  level  if  the 
liquid  flows  out  at  one  side  as  it  flows  in  at  the  other. 

One  could  even  conceive  the  bodies  J.and  B  maintaining 
the  same  temperature,  although  they  might  lose  or  gain 
certain  quantities  of  heat.  If,  for  example,  the  body  A 
were  a  mass  of  steam  ready  to  become  liquid,  and  the  body 


54  MOTIVE  POWER  OF  HEAT. 

The  operations  which  we  have  just  described 
might  have  been  performed  in  an  inverse  direction 
and  order.  There  is  nothing  to  prevent  forming 
vapor  with  the  caloric  of  the  body  B,  and  at  the 
temperature  of  that  body,  compressing  it  in  such 
a  way  as  to  make  it  acquire  the  temperature  of  the 
body  A,  finally  condensing  it  by  contact  with  this 
latter  body,  and  continuing  the  compression  to 
complete  liquefaction. 

By  our  first  operations  there  would  have  been 
at  the  same  time  production  of  motive  power 
and  transfer  of  caloric  from  the  body  A  to  the 
body  B.  By  the  inverse  operations  there  is  at  the 
same  time  expenditure  of  motive  power  and  return 
of  caloric  from  the  body  B  to  the  body  A.  But 
if  we  have  acted  in  each  case  on  the  same  quantity 
of  vapor,  if  there  is  produced  no  loss  either  of 
motive  power  or  caloric,  the  quantity  of  motive 
power  produced  in  the  first  place  will  be  equal  to 
that  which  would  have  been  expended  in  the  second, 
and  the  quantity  of  caloric  passed  in  the  first  case 
from  the  body  A  to  the  body  B  would  be  equal  to 
the  quantity  which  passes  back  again  in  the  second 
from  the  body  B  to  the  body  A ;  so  that  an  indefi- 

B  a  mass  of  ice  ready  to  melt,  these  bodies  might,  as  we 
know,  furnish  or  receive  caloric  without  thermometrig 
change. 


MOTIVE  POWER  OF  HEAT.  55 

nite  number  of  alternative  operations  of  this  sort 
could  be  carried  on  without  in  the  end  having 
either  produced  motive  power  or  transferred  caloric 
from  one  body  to  the  other. 

Now  if  there  existed  any  means  of  using  heat 
preferable  to  those  which  we  have  employed,  that 
is,  if  it  were  possible  by  any  method  whatever  to 
make  the  caloric  produce  a  quantity  of  motive 
power  greater  than  we  have  made  it  produce  by  our 
first  series  of  operations,  it  would  suffice  to  divert 
a  portion  of  this  power  in  order  by  the  method  just 
indicated  to  make  the  caloric  of  the  body  B  return 
to  the  body  A  from  the  refrigerator  to  the  furnace, 
to  restore  the  initial  conditions,  and  thus  to  be 
ready  to  commence  again  an  operation  precisely 
similar  to  the  former,  and  so  on  :  this  would  be 
not  only  perpetual  motion,  but  an  unlimited  crea- 
tion of  motive  power  without  consumption  either 
of  caloric  or  of  any  other  agent  whatever.  Such 
a  creation  is  entirely  contrary  to  ideas  now  accepted, 
to  the  laws  of  mechanics  and  of  sound  physics. 
It  is  inadmissible.*  We  should  then  conclude  that 
the  maximum  of  motive  power  resulting  from  the 
employment  of  steam  is  also  the  maximum  of  motive 
power  realizable  by  any  means  whatever.  We  will 

*  Note  A,  Appendix  B. 


56  MOTIVE  POWER  OF  HEAT. 

soon  give  a  second  more  rigorous  demonstration  of 
this  theory.  This  should  be  considered  only  as 
an  approximation.  (See  page  59.) 

We  have  a  right  to  ask,  in  regard  to  the  propo- 
sition just  enunciated,  the  following  questions: 
What  is  the  sense  of  the  word  maximum  here  ? 
By  what  sign  can  it  be  known  that  this  maximum 
is  attained  ?  By  what  sign  can  it  be  known  whether 
the  steam  is  employed  to  greatest  possible  advan- 
tage in  the  production  of  motive  power  ? 

Since  every  re-establishment  of  equilibrium  in 
the  caloric  may  be  the  cause  of  the  production  of 
motive  power,  every  re-establishment  of  equilibrium 
which  shall  be  accomplished  without  production  of 
this  power  should  be  considered  as  an  actual  loss. 
Now,  very  little  reflection  would  show  that  all 
change  of  temperature  which  is  not  due  to  a  change 
of  volume  of  the  bodies  can  be  only  a  useless  re- 
establishment  of  equilibrium  in  the  caloric.*  The 
necessary  condition  of  the  maximum  is,  then,  that 

*  We  assume  here  no  chemical  action  between  the  bodies 
employed  to  realize  the  motive  power  of  heat.  The  chem- 
ical action  which  takes  place  in  the  furnace  is,  in  some 
sort,  a  preliminary  action, — an  operation  destined  not  to 
produce  immediately  motive  power,  but  to  destroy  the 
equilibrium  of  the  caloric,  to  produce  a  difference  of  tem- 
perature which  may  finally  give  rise  to  motion. 


MOTIVE  POWER  OF  HEAT.  57 

in  the  bodies  employed  to  realize  the  motive  power 
of  heat  there  should  not  occur  any  change  of  tem- 
perature which  may  not  be  due  to  a  change  of 
volume.  Reciprocally,  every  time  that  this  condi- 
tion is  fulfilled  the  maximum  will  be  attained. 
This  principle  should  never  be  lost  sight  of  in  the 
construction  of  heat-engines  ;  it  is  its  fundamental 
basis.  If  it  cannot  be  strictly  observed,  it  should 
at  least  be  departed  from  as  little  as  possible. 

Every  change  of  temperature  which  is  not  due 
to  a  change  of  volume  or  to  chemical  action  (an 
action  that  we  provisionally  suppose  not  to  occur 
here)  is  necessarily  due  to  the  direct  passage  of  the 
caloric  from  a  more  or  less  heated  body  to  a  colder 
body.  This  passage  occurs  mainly  by  the  contact 
of  bodies  of  different  temperatures;  hence  such 
contact  should  be  avoided  as  much  as  possible.  It 
cannot  probably  be  avoided  entirely,  but  it  should 
at  least  be  so  managed  that  the  bodies  brought  in 
contact  with  each  other  differ  as  little  as  possible 
in  temperature.  When  we  just  now  supposed,  in 
our  demonstration,  the  caloric  of  the  body  A  em- 
ployed to  form  steam,  this  steam  was  considered  as 
generated  at  the  temperature  of  the  body  A  ;  thus 
the  contact  took  place  only  between  bodies  of  equal 
temperatures ;  the  change  of  temperature  occurring 
afterwards  in  the  steam  was  due  to  dilatation,  con- 


58  MOTIVE  POWER  OF  HEAT. 

sequently  to  a  change  of  volume.  Finally,  conden- 
sation took  place  also  without  contact  of  bodies  of 
different  temperatures.  It  occurred  whiJe  exert- 
ing a  constant  pressure  on  the  steam  brought  in 
contact  with  the  body  B  of  the  same  temperature 
as  itself.  The  conditions  for  a  maximum  are  thus 
found  to  be  fulfilled.  In  reality  the  operation 
cannot  proceed  exactly  as  we  have  assumed.  To 
determine  the  passage  of  caloric  from  one  body  to 
another,  it  is  necessary  that  there  should  be  an 
excess  of  temperature  in  the  first,  but  this  excess 
may  be  supposed  as  slight  as  we  please.  We  can 
regard  it  as  insensible  in  theory,  without  thereby 
destroying  the  exactness  of  the  arguments. 

A  more  substantial  objection  may  be  made  to 
our  demonstration,  thus  :  When  we  borrow  caloric 
from  the  body  A  to  produce  steam,  and  when  this 
steam  is  afterwards  condensed  by  its  contact  with 
the  body  B,  the  water  used  to  form  it,  and  which 
we  considered  at  first  as  being  of  the  temperature 
of  the  body  A,  is  found  at  the  close  of  the  opera- 
tion at  the  temperature  of  the  body  B.  It  has 
become  cool.  If  we  wish  to  begin  again  an  opera- 
tion similar  to  the  first,  if  we  wish  to  develop  a 
new  quantity  of  motive  power  with  the  same  in- 
strument, with  the  same  steam,  it  is  necessary  first 
to  re-establish  the  original  condition — to  restore 


MOTIVE  POWER  OF  HEAT.  59 

the  water  to  the  original  temperature.  This  can 
undoubtedly  be  done  by  at  once  putting  it  again 
in  contact  with  the  body  A  ;  but  there  is  then 
contact  between  bodies  of  different  temperatures, 
and  loss  of  motive  power.*  It  would  be  impossi- 
ble to  execute  the  inverse  operation,  that  is,  to 
return  to  the  body  A  the  caloric  employed  to  raise 
the  temperature  of  the  liquid. 

This  difficulty  may  be  removed  by  supposing  the 
difference  of  temperature  between  the  body  A  and 
the  body  B  indefinitely  small.  The  quantity  of 
heat  necessary  to  raise  the  liquid  to  its  former 

*  This  kind  of  loss  is  found  in  all  steam-engines.  In 
fact,  the  water  destined  to  feed  the  boiler  is  always  cooler 
than  the  water  which  it  already  contains.  There  occurs 
between  them  a  useless  re-establishment  of  equilibrium  of 
caloric.  We  are  easily  convinced,  a  posteriori,  that  this  re- 
establishment  of  equilibrium  causes  a  loss  of  motive  power 
if  we  reflect  that  it  would  have  been  possible  to  previously 
heat  the  feed-water  by  using  it  as  condensing-water  in  a 
small  accessory  engine,  when  the  steam  drawn  from  the 
large  boiler  might  have  been  used,  and  where  the  conden- 
sation might  be  produced  at.  a  temperature  intermediate 
between  that  of  the  boiler  and  that  of  the  principal  con- 
denser. The  power  produced  by  the  small  engine  would 
have  cost  no  loss  of  heat,  since  all  that  which  had  been 
used  would  have  returned  into  the  boiler  with  the  water  of 
condensation. 


60  MOTIVE  POWER  OF  HEAT. 

temperature  will  be  also  indefinitely  small  and  un- 
important relatively  to  that  which  is  necessary  to 
produce  steam — a  quantity  always  limited. 

The  proposition  found  elsewhere  demonstrated 
for  the  case  in  which  the  difference  between  the 
temperatures  of  the  two  bodies  is  indefinitely  small, 
may  be  easily  extended  to  the  general  case.  In 
fact,  if  it  operated  to  produce  motive  power  by  the 
passage  of  caloric  from  the  body  A  to  the  body  Z, 
the  temperature  of  this  latter  body  being  very  dif- 
ferent from  that  of  the  former,  we  should  imagine 
a  series  of  bodies  B,  C,  D .  .  .  of  temperatures 
intermediate  between  those  of  the  bodies  A,  Z, 
and  selected  so  that  the  differences  from  A  to  B, 
from  B  to  C,  etc.,  may  all  be  indefinitely  small. 
The  caloric  coming  from  A  would  not  arrive  at  Z 
till  after  it  had  passed  through  the  bodies  B,  C,  D, 
etc.,  and  after  having  developed  in  each  of  these 
stages  maximum  motive  power.  The  inverse 
operations  would  here  be  entirely  possible,  and  the 
reasoning  of  page  52  would  be  strictly  applicable. 

According  to  established  principles  at  the  present 
time,  we  can  compare  with  sufficient  accuracy  the 
motive  power  of  heat  to  that  of  a  waterfall.  Each 
has  a  maximum  that  we  cannot  exceed,  whatever 
may  be,  on  the  one  hand,  the  machine  which  is 
acted  upon  by  the  water,  and  whatever,  on  the 


MOTIVE  POWER  OF  HEAT.  61 

other  hand,  the  substance  acted  upon  by  the  heat. 
The  motive  power  of  a  waterfall  depends  on  its 
height  and  on  the  quantity  of  the  liquid;  the 
motive  power  of  heat  depends  also  on  the  quantity 
of  caloric  used,  and  on  what  may  be  termed,  on 
what  in  fact  we  will  call,  the  height  of  its  fall,* 
that  is  to  say,  the  difference  of  temperature  of  the 
bodies  between  which  the  exchange  of  caloric  is 
made.  In  the  waterfall  the  motive  power  is  ex- 
actly proportional  to  the  difference  of  level  between 
the  higher  and  lower  reservoirs.  In  the  fall  of 
caloric  the  motive  power  undoubtedly  increases 
with  the  difference  of  temperature  between  the 
warm  and  the  cold  bodies ;  but  we  do  not  know 
whether  it  is  proportional  to  this  difference.  We 
do  not  know,  for  example,  whether  the  fall  of  ca- 
loric from  100  to  50  degrees  furnishes  more  or  less 
motive  power  than  the  fall  of  this  same  caloric  from 
50  to  zero.  It  is  a  question  which  we  propose  to 
examine  hereafter. 

We  shall  give  here  a  second  demonstration  of 
the  fundamental  proposition  enunciated  on  page 
56,  and  present  this  proposition  under  a  more  gen- 
eral form  than  the  one  already  given. 

*  The  matter  here  dealt  with  being  entirely  new,  we  are 
obliged  to  employ  expressions  not  in  use  as  yet,  and  which 
perhaps  are  less  clear  than  is  desirable. 


62  MOTIVE  POWER  OF  HEAT. 

When  a  gaseous  fluid  is  rapidly  compressed  its 
temperature  rises.  It  falls,  on  the  contrary,  when 
it  is  rapidly  dilated.  This  is  one  of  the  facts  best 
demonstrated  by  experiment.  We  will  take  it  for 
the  basis  of  our  demonstration.* 

If,  when  the  temperature  of  a  gas  has  been 
raised  by  compression,  we  wish  to  reduce  it  to  its 
former  temperature  without  subjecting  its  volume 
to  new  changes,  some  of  its  caloric  must  be  re- 
moved. This  caloric  might  have  been  removed  in 
proportion  as  pressure  was  applied,  so  that  the 
temperature  of  the  gas  would  remain  constant. 
Similarly,  if  the  gas  is  rarefied  we  can  avoid  lower- 
ing the  temperature  by  supplying  it  with  a  cer- 
tain quantity  of  caloric.  Let  us  call  the  caloric 
employed  at  such  times,  when  no  change  of  tem- 
perature occurs,  caloric  due  to  change  of  volume. 
This  denomination  does  not  indicate  that  the 
caloric  appertains  to  the  volume  :  it  does  not  ap- 
pertain to  it  any  more  than  to  pressure,  and 
might  as  well  be  called  caloric  due  to  the  change 
of  pressure.  We  do  not  know  what  laws  it 
follows  relative  to  the  variations  of  volume  :  it  is 
possible  that  its  quantity  changes  either  with  the 
nature  of  the  gas,  its  density, or  its  temperature.  Ex- 

*  Note  13,  Appendix  B. 


MOTIVE  POWER  OF  HEAT. 


63 


periment  has  taught  us  nothing  on  this  subject.  It 
has  only  shown  us  that  this  caloric  is  developed  in 
greater  or  less  quantity  by  the  compression  of  the 
elastic  fluids. 

This  preliminary  idea  being  established,  let  us 
imagine  an  elastic  fluid,  atmospheric  air  for  exam- 
ple, shut  up  in  a  cylindrical  vessel,  abed.  (Fig.  1), 
provided  with  a  movable  dia- 
phragm or  piston,  cd.  Let 
there  be  also  two  bodies,  A  and 
B,  kept  each  at  a  constant 
temperature,  that  of  A  being 
higher  than  that  of  B.  Let 
us  picture  to  ourselves  now 
the  series  of  operations  which 
are  to  be  described  : 

(1)  Contact     of    the    body 
A  with  the  air  enclosed  in  the 
space  abed    or  with   the   wall 
of  this  space — a  wall  that  we 
will    suppose  to    transmit  the 
caloric   readily.     The   air    be- 
comes by  such  contact  of  the 

same  temperature  as  the  body  A\  cd  is  the  actual 
position  of  the  piston. 

(2)  The  piston  gradually  rises  and  takes  the 
position  ef.    The  body  A  is  all  the  time  in  con- 


FlG.  1 


64  MOTIVE  POWER  OF  HEAT. 

tact  with  the  air,  which  is  thus  kept  at  a  constant 
temperature  during  the  rarefaction.  The  body  A 
furnishes  the  caloric  necessary  to  keep  the  tem- 
perature constant. 

(3)  The  body  A  is  removed,  and  the  air  is  then 
no  longer  in  contact  with  any  body  capable  of  fur- 
nishing  it  with   caloric.     The   piston  meanwhile 
continues  to  move,  and  passes  from  the  position  ef 
to  the   position  gh.     The  air   is  rarefied  without 
receiving  caloric,  and  its  temperature  falls.     Let 
us  imagine  that  it  falls  thus  till  it  becomes  equal 
to  that  of  the  body  B\  at  this  instant  the  piston 
stops,  remaining  at  the  position  gh. 

(4)  The  air  is  placed  in  contact  with  the  body 
B\  it  is  compressed  by  the  return  of  the  piston  as 
it  is  moved  from  the  position  gh  to  the  position 
cd.     This  air  remains,   however,  at    a    constant 
temperature  because  of  its  contact  with  the ,  body 
B9  to  which  it  yields  its  caloric. 

(5)  The  body  B  is  removed,  and  the  compres- 
sion of  the  air   is  continued,  which  being  then 
isolated,  its  temperature  rises.     The  compression 
is  continued  till  the  air  acquires  the  temperature 
of  the  body  A.     The  piston   passes  during  this 
time  from  the  position  cd  to  the  position  ik. 

(6)  The  air  is  again  placed  in  contact  with  the 
body  A.     The  piston  returns  from  the  position  iJc 


MOTIVE  POWER  OF  HEAT.  65 

to  the  position  ef ;  the  temperature  remains  un- 
changed. 

(7)  The  step  described  under  number  3  is  re- 
newed, then  successively  the  steps  4,  5,  6,  3,  4,  5, 
6,  3,  4,  5  ;  and  so  on. 

In  these  various  operations  the  piston  is  subject 
to  an  effort  of  greater  or  less  magnitude,  exerted 
by  the  air  enclosed  in  the  cylinder;  the  elastic 
force  of  this  air  varies  as  much  by  reason  of  the 
changes  in  volume  as  of  changes  of  temperature. 
But  it  should  -be  remarked  that  with  equal 
volumes,  that  is,  for  the  similar  positions  of  the 
piston,  the  temperature  is  higher  during  the  move- 
ments of  dilatation  than  during  the  movements  of 
compression.  During  the  former  the  elastic  force 
of  the  air  is  found  to  be  greater,  and  consequently 
the  quantity  of  motive  power  produced  by  the 
movements  of  dilatation  is  more  considerable  than 
that  consumed  to  produce  the  movements  of  com- 
pression.. Thus  we  should  obtain  an  excess  of 
motive  power — an  excess  which  we  could  employ 
for  any  purpose  whatever.  The  air,  then,  has 
served  as  a  heat-engine  ;  we  have,  in  fact,  employed 
it  in  the  most  advantageous  manner  possible,  for 
no  useless  re-establishment  of  equilibrium  has 
been  effected  in  the  caloric. 

All    the    above-described    operations    may    be 


66  MOTIVE  POWER  OF  HEAT. 

executed  in  an  inverse  sense  and  order.  Let  us 
imagine  that,  after  the  sixth  period,  that  is  to  say 
the  piston  having  arrived  at  the  position  ef,  we 
cause  it  to  return  to  the  position  ik,  and  that  at 
the  same  time  we  keep  the  air  in  contact  with  the 
body  A.  The  caloric  furnished  by  this  body 
during  the  sixth  period  would  return  to  its  source, 
that  is,  to  the  body  A,  and  the  conditions  would 
then  become  precisely  the  same  as  they  were  at  the 
end  of  the  fifth  period.  If  now  we  take  away  the 
body  A,  and  if  we  cause  the  piston  to  move  from 
ef  to  cd,  the  temperature  of  the  air  will  diminish 
as  many  degrees  as  it  increased  during  the  fifth 
period,  and  will  become  that  of  the  body  B.  We 
may  evidently  continue  a  series  of  operations  the 
inverse  of  those  already  described.  It  is  only 
necessary  under  the  same  circumstances  to  exe- 
cute for  each  period  a  movement  of  dilatation 
instead  of  a  movement  of  compression,  and  re- 
ciprocally. 

The  result  of  these  first  operations  has  been  the 
production  of  a  certain  quantity  of  motive  power 
and  the  removal  of  caloric  from  the  body  A  to  the 
body  B.  The  result  of  the  inverse  operations  is 
the  consumption  of  the  motive  power  produced  and 
the  return  of  the  caloric  from  the  body  B  to  the 
body  A ;  so  that  these  two  series  of  operations  annul 


MOTIVE  POWER  OF  HEAT.  67 

each  other,  after  a  fashion,  one  neutralizing  the 
other. 

The  impossibility  of  making  the  caloric  produce 
a  greater  quantity  of  motive  power  than  that  which 
we  obtained  from  it  by  our  first  series  of  opera- 
tions, is  now  easily  proved.  It  is  demonstrated  by 
reasoning  very  similar  to  that  employed  at  page  5G; 
the  reasoning  will  here  be  even  more  exact.  The 
air  which  we  have  used  to  develop  the  motive 
power  is  restored  at  the  end  of  each  cycle  of  opera- 
tions exactly  to  the  state  in  which  it  was  at  first 
found,  while,  as  we  have  already  remarked,  this 
would  not  be  precisely  the  case  with  the  vapor  of 
water.* 

*  "We  tacitly  assume  in  our  demonstration,  that  when  a 
body  has  experienced  any  changes,  and  when  after  a  cer- 
tain number  of  transformations  it  returns  to  precisely  its 
original  state,  that  is,  to  that  state  considered  in  respect  to 
density,  to  temperature,  to  mode  of  aggregation — let  us 
suppose,  I  say,  that  this  body  is  found  to  contain  the  same 
quantity  of  heat  that  it  contained  at  first,  or  else  that  the 
quantities  of  heat  absorbed  or  set  free  in  these  different 
transformations  are  exactly  compensated.  This  fact  has 
never  been  culled  in  question.  It  was  first  admitted  with- 
out reflection,  and  verified  afterwards  in  many  cases  by 
experiments  with  the  calorimeter.  To  deny  it  would  be 
to  overthrow  the  whole  theory  of  heat  to  which  it  serves 
as  a  basis.  For  the  rest,  we  may  say  in  passing,  the  main 


68  MOTIVE  POWER  OF  HEAT. 

We  have  chosen  atmospheric  air  as  the  instru- 
ment which  should  develop  the  motive  power  of 
heat,  but  it  is  evident  that  the  reasoning  would 
have  heen  the  same  for  all  other  gaseous  substances, 
and  even  for  all  other  bodies  susceptible  of  change 
of  temperature  through  successive  contractions  and 
dilatations,  which  comprehends  all  natural  sub- 
stances, or  at  least  all  those  which  are  adapted  to 
realize  the  motive  power  of  heat.  Thus  we  are  led 
to  establish  this  general  proposition : 

The  motive  power  of  heat  is  independent  of  the 
agents  employed  to  realize  it ;  its  quantity  is  fixed 
solely  by  the  temperatures  of  the  bodies  between 
which  is  effected,  finally,  the  transfer  of  the  caloric. 

We  must  understand  here  that  each  of  the 
methods  of  developing  motive  power  attains  the 
perfection  of  which  it  is  susceptible.  This  condi- 
tion is  found  to  be  fulfilled  if,  as  we  remarked 
above,  there  is  produced  in  the  body  no  other 
change  of  temperature  than  that  due  to  change  of 
volume,  or,  what  is  the  same  thing  in  other  words, 
if  there  is  no  contact  between  bodies  of  sensibly 
different  temperatures. 

Different  methods  of  realizing  motive  power  may 

principles  on  which  the  theory  of  heat  rests  require  the 
most  careful  examination.  Many  experimental  facts  ap- 
pear almost  inexplicable  in  the  present  state  of  this  theory. 


MOTIVE  POWER  OF  HEAT.  69 

be  taken,  as  in  the  employment  of  different  sub- 
stances, or  in  the  use  of  the  same  substance  in  two 
different  states — for  example,  of  a  gas  at  two  dif- 
ferent densities. 

This  leads  us  naturally  to  those  interesting  re- 
searches on  the  aeriform  fluids — researches  which 
lead  us  also  to  new  results  in  regard  to  the  motive 
power  of  heat,  and  give  us  the  means  of  verifying, 
in  some  particular  cases,  the  fundamental  proposi- 
tion above  stated.* 

We  readily  see  that  our  demonstration  would 
have  been  simplified  by  supposing  the  temperatures 
of  the  bodies  A  and  B  to  differ  very  little.  Then 
the  movements  of  the  piston  being  slight  during 
the  periods  3  and  5,  these  periods  might  have  been 
suppressed  without  influencing  sensibly  the  pro- 
duction of  motive  power.  A  very  little  change  of 
volume  should  suffice  in  fact  to  produce  a  very 
slight  change  of  temperature,  and  this  slight  change 
of  volume  may  be  neglected  in  presence  of  that  of 
the  periods  4  and  6,  of  which  the  extent  is  unlim- 
ited. 

If  we  suppress  periods  3  and  5,  in  the  series  of 

*  We  will  suppose,  in  what  follows,  the  reader  to  be  au 
courant  with  the  later  progress  of  modern  Physics  in  re- 
gard to  gaseous  substances  and  heat. 


70 


MOTIVE  POWER  OF  HEAT. 


operations  above  described,  it  is  reduced  to  the  fol- 
lowing : 

(1)  Contact  of  the  gas  confined  in  abed  (Fig.  2) 
with  the  body  A,  passage  of  the  piston  from  cd  to  ef. 


— 

f_  e[ 

_— 



d            c' 

I 

FlQ 

1 

.  2. 

>                <• 

(/ 

FlQ 

6 

.  3. 

(2)  Eemoval  of  the  body  A,  contact  of  the  gas 
confined  in  abef  with  the  body  B,  return  of  the 
piston  from  efto  cd. 

(3)  Removal  of  the  body  B,  contact  of  the  gas 
with  the  body  A,  passage  of  the  piston  from  cd  to 
ef,  that  is,  repetition  of  the  first  period,  and  so  on. 

The  motive  power  resulting  from  the  ensemble 
of  operations  1  and  2  will  evidently  be  the  differ- 
ence between  that  which  is  produced  by  the  expan- 
sion of  the  gas  while  it  is  at  the  temperature  of  the 
body  A,  and  that  which  is  consumed  to  compress 
this  gas  while  it  is  at  the  temperature  of  the 
body  B. 


MOTIVE  POWER  OF  HEAT.  71 

Let  us  suppose  that  operations  1  and  2  be  per- 
formed on  two  gases  of  different  chemical  natures 
but  under  the  same  pressure — under  atmospheric 
pressure,  for  example.  These  two  gases  will  be- 
have exactly  alike  under  the  same  circumstances, 
that  is,  their  expansive  forces,  originally  equal, 
will  remain  always  equal,  whatever  may  be  the 
variations  of  volume  and  of  temperature,  provided 
these  variations  are  the  same  in  both.  This  results 
obviously  from  the  laws  of  Mariotte  and  MM.  Gay- 
Lussac  and  Dalton — laws  common  to  all  elastic 
fluids,  and  in  virtue  of  which  the  same  relations 
exist  for  all  these  fluids  between  the  volume,  the 
expansive  force,  and  the  temperature. 

Since  two  different  gases  at  the  same  tempera- 
ture and  under  the  same  pressure  should  behave 
alike  under  the  same  circumstances,  if  we  subjected 
them  both  to  the  operations  above  described,  they 
should  give  rise  to  equal  quantities  of  motive  power. 

Now  this  implies,  according  to  the  fundamental 
proposition  that  we  have  established,  the  employ- 
ment of  two  equal  quantities  of  caloric;  that  is,  it 
implies  that  the  quantity  of  caloric  transferred  from 
the  body  A  to  the  body  B  is  the  same,  whichever 
gas  is  used. 

The  quantity  of  caloric  transferred  from  the 
body  A  to  the  body  B  is  evidently  that  which  is 


72  MOTIVE  POWER  OF  HEAT. 

absorbed  by  the  gas  in  its  expansion  of  volume,  or 
that  which  this  gas  relinquishes  during  compres- 
sion. We  are  led,  then,  to  establish  the  following 
proposition  : 

When  a  gas  passes  without  change  of  tempera- 
ture from  one  definite  volume  and  pressure  to  an- 
other volume  and  another  pressure  equally  definite, 
the  quantity  of  caloric  absorbed  or  relinquished  is 
always  the  same,  ivhatever  may  be  the  nature  of 
the  gas  chosen  as  the  subject  of  the  experiment. 

Take,  for  example,  1  litre  of  air  at  the  tempera- 
ture of  100°  and  under  the  pressure  of  one  atmos- 
phere. If  we  double  the  volume  of  this  air  and 
wish  to  maintain  it  at  the  temperature  of  100°,  a 
certain  quantity  of  heat  must  be  supplied  to  it. 
Now  this  quantity  will  be  precisely  the  same  if, 
instead  of  operating  on  the  air,  we  operate  upon 
carbonic-acid  gas,  upon  nitrogen,  upon  hydrogen, 
upon  vapor  of  water  or  of  alcohol,  that  is,  if  we 
double  the  volume  of  1  litre  of  these  gases  taken  at 
the  temperature  of  100°  and  under  atmospheric 
pressure. 

It  will  be  the  same  thing  in  the  inverse  sense  if, 
Instead  of  doubling  the  volume  of  gas,  we  reduce 
it  one  half  by  compression.  The  quantity  of  heat 
that  the  elastic  fluids  set  free  or  absorb  in  their 
changes  of  volume  has  never  been  measured  by 


MOTIVE  POWER  OF  HEAT,  73 

any  direct  experiment,  and  doubtless  such  an  ex- 
periment would  be  very  difficult,  but  there  exists  a 
datum  which  is  very  nearly  its  equivalent.  This 
has  been  furnished  by  the  theory  of  sound.  It  de- 
serves much  confidence  because  of  the  exactness  of 
the  conditions  which  have  led  to  its  establishment. 
It  consists  in  this : 

Atmospheric  air  should  rise  one  degree  Centi- 
grade when  by  sudden  compression  it  experiences 
a  reduction  of  volume  of  Tfg-.* 

Experiments  on  the  velocity  of  sound  having 
been  made  in  air  under  the  pressure  of  760  milli- 
metres of  mercury  and  at  the  temperature  of  6°, 
it  is  only  to  these  two  circumstances  that  our 
datum  has  reference.  We  will,  however,  for  greater 
facility,  refer  it  to  the  temperature  0°,  which  is 
nearly  the  same. 

Air  compressed  Tfg-,  and  thus  heated  one  degree, 
differs  from  air  heated  directly  one  degree  only  in 
its  density.  The  primitive  volume  being  supposed 


*  M.  Poisson,  to  whom  this  figure  is  due,  has  shown 
that  it  accords  very  well  with  the  result  of  an  experiment 
of  MM.  Clement  and  Desormes  on  the  return  of  air  into  a 
vacuum,  or  rather,  into  air  slightly  rarefied.  It  also  ac- 
cords very  nearly  with  results  found  by  MM.  Gay-Lussaq 
and  Welter.  (See  note,  p.  87.) 


74  MOTIVE  POWER  OF  HEAT. 

to  be    V,  the  compression   of  TTT  reduces  it  to 
V-j^V. 

Direct  heating  under  constant  pressure  should, 
according  to  the  rule  of  M.  Gay-Lussac,  increase 
the  volume  of  air  ¥£T  above  what  it  would  be  at  0°  : 
so  the  air  is,  on  the  one  hand,  reduced  to  the  vol- 
ume V  —  TT7  F;  on  the  other,  it  is  increased  to 


_ 

The  difference  between  the  quantities  of  heat 
which  the  air  possesses  in  both  cases  is  evidently 
the  quantity  employed  to  raise  it  directly  one  de- 
gree; so  then  the  quantity  of  heat  that  the  air 
would  absorb  in  passing  from  the  volume  V  --  T}T  V 
to  the  volume  F  -\-  ^V  is  equal  to  that  which 
is  required  to  raise  it  one  degree. 

Let  us  suppose  now  that,  instead  of  heating  one 
degree  the  air  subjected  to  a  constant  pressure  and 
able  to  dilate  freely,  we  inclose  it  within  an  invari- 
able space,  and  that  in  this  condition  we  cause  it 
to  rise  one  degree  in  temperature.  The  air  thus 
heated  one  degree  will  differ  from  the  air  com- 
pressed TT^-  only  by  its  1T-g-  greater  volume.  So 
then  the  quantity  of  heat  that  the  air  would  set 
free  by  a  reduction  of  volume  of  yir  is  equal  to 
that  which  would  be  required  to  raise  it  one  degree 
Centigrade  under  constant  volume.  As  the  differ- 
ences between  the  volumes  F  —  T|-g  F,  F,  and 


•MOTIVE  POWER  OF  HEAT.  75 

V  -f-  ¥JT  V  are  small  relatively  to  the  volumes 
themselves,  we  may  regard  the  quantities  of  heat 
absorbed  by  the  air  in  passing  from  the  first  of 
these  volumes  to  the  second,  and  from  the  first  to 
the  third,  as  sensibly  proportional  to  the  changes 
of  volume.  We  are  then  led  to  the  establishment 
of  the  following  relation : 

The  quantity  of  heat  necessary  to  raise  one  de- 
gree air  under  constant  pressure  is  to  the  quantity 
of  heat  necessary  to  raise  one  degree  the  same  air 
under  constant  volume,  in  the  ratio  of  the  numbers 

rhr  +  irh-    to    TIT; 

or,  multiplying  both  by  116  X  267,  in  the  ratio  of 
the  numbers  267  +  116  to  267. 

This,  then,  is  the  ratio  which  exists  between  the 
capacity  of  air  for  heat  under  constant  pressure 
and  its  capacity  under  constant  volume.  If  the 
first  of  these  two  capacities  is  expressed  by  unity, 
the  other  will  be  expressed  by  the  number  267+7116 , 
or  very  nearly  0.700;  their  difference,  1  —  0.700  or 
0.300,  will  evidently  express  the  quantity  of  heat 
which  will  produce  the  increase  of  volume  in  the 
air  when  it  is  heated  one  degree  under  constant 
pressure. 

According  to  the  law  of  MM.  Gay-Lussac  and 
JDalton,  this  increase  of  volume  would  be  the  same 


76 


MOTIVE  POWER  OF  HEAT. 


for  all  other  gases;  according  to  the  theory  demon- 
strated on  page  87,  the  heat  absorbed  by  these  equal 
increases  of  volume  is  the  same  for  all  the  elastic 
fluids,  which  leads  to  the  establishment  of  the  fol- 
lowing proposition : 

The  difference  between  specific  heat  under  con- 
stant pressure  and  specific  heat  under  constant 
volume  is  the  same  for  all  gases. 

It  should  be  remarked  here  that  all  the  gases 
are  considered  as  taken  under  the  same  pressure, 
atmospheric  pressure  for  example,  and  that  the 
specific  heats  are  also  measured  with  reference  to 
the  volumes. 

It  is  a  very  easy  matter  now  for  us  to  prepare  a 
table  of  the  specific  heat  of  gases  under  constant 
volume,  from  the  knowledge  of  their  specific  heats 
under  constant  pressure.     Here  is  the  table : 
TABLE  OF  THE  SPECIFIC  HEAT  OF  GASES. 


NAMES  OF  GASES. 

Specific  Heat 
under 
Const.  Press. 

Specific  Heat 
at 
Const.  Vol. 

Atmospheric  Air,    .... 

1.000 

0.700 

Hydrogen  Gas,    
Carbonic  Acid,  

0.903 
1.258 

0.603 
0.958 

0.976 

0.676 

Nitrosren                                 . 

1  000 

0  700 

Protoxide  of  Nitrogen,     .     . 
Olefiant  Gas                      .     . 

1.350 
1.553 

1.050 
1.253 

Oxide  of  Carbon,     .... 

1.034 

0.734 

MOTIVE  POWER  OF  HEAT.  77 

The  first  column  is  the  result  of  the  direct 
experiments  of  MM.  Delaroche  and  Berard  on  the 
specific  heat  of  the  gas  under  atmospheric  pressure, 
and  the  second  column  is  composed  of  the  numbers 
of  the  first  diminished  by  0.300. 

The  numbers  of  the  first  column  and  those  of 
the  second  are  here  referred  to  the  same  unit,  to 
the  specific  heat  of  atmospheric  air  under  constant 
pressure. 

The  difference  between  each  number  of  the  first 
column  and  the  corresponding  number  of  the  sec- 
ond being  constant,  the  relation  between  these 
numbers  should  be  variable.  Thus  the  relation 
between  the  specific  heat  of  gases  under  constant 
pressure  and  the  specific  heat  at  constant  volume, 
varies  in  different  gases. 

We  have  seen  that  air  when  it  is  subjected  to  a 
sudden  compression  of  Tfg-  of  its  volume  rises  one 
degree  in  temperature.  The  other  gases  through 
a  similar  compression  should  also  rise  in  tempera- 
ture. They  should  rise,  but  not  equally,  in  inverse 
ratio  with  their  specific  heat  at  constant  volume. 
In  fact,  the  reduction  of  volume  being  by  hypothe- 
sis always  the  same,  the  quantity  of  heat  due  to 
this  reduction  should  likewise  be  always  the  same, 
and  consequently  should  produce  an  elevation  of 
temperature  dependent  only  on  the  specific  heat 


78  MOTIVE  POWER  OF  HEAT. 

acquired  by  the  gas  after  its  compression,  and 
evidently  in  inverse  ratio  with  this  specific  heat. 
Thus  we  can  easily  form  the  table  of  the  elevations 
of  temperature  of  the  different  gases  for  a  compres- 
sion of  yfg-. 

TABLE  OF  THE  ELEVATION  OP  TEMPERATURE 

OF 

Oases  through  the  Effect  of  Compression. 


NAMES  OP  GASES. 

Elevation  of  Temperature 
for  a  Reduction  of 
Volume  of  y^. 

1.000 

1.160 

0.730 

1.035 

Nitrogen,      

1.000 

Protoxide  of  Nitrogen,  .     .    . 
Olefiant  Gas      ...... 

0.667 
0.558 

0.955 

A  second  compression  of  T|^-  (of  the  altered  vol- 
ume), as  we  shall  presently  see,  would  also  raise  the 
temperature  of  these  gases  nearly  as  much  as  the 
first;  but  it  would  not  be  the  same  with  a  third,  a 
fourth,  a  hundredth  such  compression.  The  capac- 
ity of  gases  for  heat  changes  with  their  volume. 
It  is  not  unlikely  that  it  changes  also  with  the 
temperature. 

We  shall  now  deduce  from  the  general  proposi- 


MOTIVE  POWER  OF  HEAT.  79 

tion  stated  on  page  68  a  second  theory,  which  will 
serve  as  a  corollary  to  that  just  demonstrated. 

Let  us  suppose  that  the  gas  enclosed  in  the 
cylindrical  space  abed  (Fig.  2)  be  transported  into 
the  space  a'b'c'd'  (Fig.  3)  of  equal  height,  but  of 
different  base  and  wider.  This  gas  would  increase 
in  volume,  would  diminish  in  density  and  in  elastic 
force,  in  the  inverse  ratio  of  the  two  volumes  abed, 
a'b'c'd'.  As  to  the  total  pressure  exerted  in  each 
piston  cd,  c'd',  it  would  be  the  same  from  all  quar- 
ters, for  the  surface  of  these  pistons  is  in  direct 
ratio  to  the  volumes. 

Let  us  suppose  that  we  perform  on  the  gas  in- 
closed in  a'b'c'd'  the  operations  described  on  page 
70,  and  which  were  taken  as  having  been  performed 
upon  the  gas  inclosed  in  abed',  that  is,  let  us  sup- 
pose that  we  have  given  to  the  piston  c'd'  motions 
equal  to  those  of  the  piston  cd,  that  we  have  made 
it  occupy  successively  the  positions  c'd'  correspond- 
ing to  cd,  and  e'f  corresponding  to  ef,  and  that  at 
the  same  time  we  have  subjected  the  gas  by  means 
of  the  two  bodies  A  and  B  to  the  same  variations 
of  temperature  as  when  it  was  inclosed  in  abed 
The  total  effort  exercised  on  the  piston  would  be 
found  to  be,  in  the  two  cases,  always  the  same  at 
the  corresponding  instants.  This  results  solely  from 


Of     r> 


80  MOTIVE  POWER  OF  SEAT. 

the  l&w  <vf  Mariotte.*  In  fact,  the  densities  of  the 
two  gases  maintaining  always  the  same  ratio  for 
similar  positions  of  the  pistons,  and  the  tempera- 
tures being  always  equal  in  both,  the  total  pressures 
exercised  on  the  pistons  will  always  maintain  the 
same  ratio  to  each  other.  If  this  ratio  is,  at  any 
instant  whatever,  unity,  the  pressures  will  always 
be  equal. 

As,  furthermore,  the  movements  of  the  two  pis- 
tons have  equal  extent,  the  motive  power  produced 
by  each  will  evidently  be  the  same;  whence  we 
should  conclude,  according  to  the  proposition  on 

*  The  law  of  Mariotte,  which  is  here  made  the  founda- 
tion upon  which  to  establish  our  demonstration,  is  one  of 
the  best  authenticated  physical  laws.  It  has  served  as  a 
basis  to  many  theories  verified  by  experience,  and  which 
in  turn  verify  all  the  laws  on  which  they  are  founded. 
We  can  cite  also,  as  a  valuable  verification  of  Mariotte's 
law  and  also  of  that  of  MM.  Gay-Lussac  and  Dalton,  for  a 
great  difference  of  temperature,  the  experiments  of  MM. 
Dulong  and  Petit.  (See  Annales  de  CMmie  el  de  Physique, 
Feb.  1818,  t.  vii.  p.  122.) 

The  more  recent  experiments  of  Davy  and  Faraday  can 
also  be  cited. 

The  theories  that  we  deduce  here  would  not  perhaps  be 
exact  if  applied  outside  of  certain  limits  either  of  density 
or  temperature.  They  should  be  regarded  as  true  only 
within  the  limits  in  which  the  laws  of  Mariotte  and  of 
MM.  Gay-Lussac  and  Dalton  are  themselves  proven. 


MOTIVE  POWER  OF  HEAT.  81 

page  68,  that  the  quantities  of  heat  consumed  by 
each  are  the  same,  that  is,  that  there  passes  from 
the  body  A  to  the  body  B  the  same  quantity  of 
heat  in  both  cases. 

The  heat  abstracted  from  the  body  A  and  com- 
municated to  the  body  B,  is  simply  the  heat  ab- 
sorbed during  the  rarefaction  of  the  gas,  and  after- 
wards liberated  by  its  compression.  We  are  therefore 
led  to  establish  the  following  theorem : 

When  an  elastic  fluid  passes  without  change  of 
temperature  from  the  volume  U  to  the  volume  V, 
and  when  a  similar  ponderable  quantity  of  the 
same  gas  passes  at  the  same  temperature  from  the 
volume  V  to  the  volume  V,  if  the  ratio  of  U'  to 
V  is  found  to  be  the  same  as  the  ratio  of  U  to  V, 
the  quantities  of  heat  absorbed  or  disengaged  in 
the  two  cases  will  be  equal. 

This  theorem  might  also  be  expressed  as  follows : 

When  a  gas  varies  in  volume  without  change  of 
temperature,  the  quantities  of  heat  absorbed  or 
liberated  by  this  gas  are  in  arithmetical  progres- 
sion, if  the  increments  or  the  decrements  of  volume 
are  found  to  be  in  geometrical  progression. 

When  a  litre  of  air  maintained  at  a  temperature 
of  ten  degrees  is  compressed,  and  when  it  is  re- 
duced to  one  half  a  litre,  a  certain  quantity  of 
heat  is  set  free.  This  quantity  will  be  found  always 


32  MOTIVE  POWER  OF  HEAT. 

the  same  if  the  volume  is  further  reduced  from  a 
half  litre  to  a  quarter  litre,  from  a  quarter  litre  to 
an  eighth,  and  so  on. 

If,  instead  of  compressing  the  air,  we  carry  it 
successively  to  two  litres,  four  litres,  eight  litres, 
etc.,  it  will  be  necessary  to  supply  to  it  always  equal 
quantities  of  heat  in  order  to  maintain  a  constant 
temperature. 

This  readily  accounts  for  the  high  temperature 
attained  by  air  when  rapidly  compressed.  We 
know  that  this  temperature  inflames  tinder  and 
even  makes  air  luminous.  If,  for  a  moment,  we 
suppose  the  specific  heat  of  air  to  be  constant,  in 
spite  of  the  changes  of  volume  and  temperature, 
the  temperature  will  increase  in  arithmetical  pro- 
gression for  reduction  of  volume  in  geometrical 
progression. 

Starting  from  this  datum,  and  admitting  that 
one  degree  of  elevation  in  the  temperature  cor- 
responds to  a  compression  of  T-\-¥,  we  shall  readily 
come  to  the  conclusion  that  air  reduced  to  -fa  of 
its  primitive  volume  should  rise  in  temperature 
about  300  degrees,  which  is  sufficient  to  inflame 
tinder.* 

*  When  the  volume  is  reduced  TT^,  that  is,  when  it 
becomes  yyf  of  what  it  was  at  first,  the  temperature  rises 
one  degree.  Another  reduction  of  TT^  carries  the  volume 


MOTIVE  POWER  OF  HEAT.  83 

The  elevation  of  temperature  ought,  evidently, 
to  be  still  more  considerable  if  the  capacity  of  the 
air  for  heat  becomes  less  as  its  volume  diminishes. 
Now  this  is  probable,  and  it  also  seems  to  follow 
from  the  experiments  of  MM.  Delaroche  and 
Berard  on  the  specific  heat  of  air  taken  at  different 
densities.  (See  the  Memoire  in  the  Annales  de 
Chimie,  t.  Ixxxv.  pp.  72,  224.) 

The  two  theorems  explained  on  pp.  72  and  81 
suffice  for  the  comparison  of  the  quantities  of  heat 
absorbed  or  set  free  in  the  changes  of  volume  of 
elastic  fluids,  whatever  may  be  the  density  and  the 
chemical  nature  of  these  fluids,  provided  always 

to  (Hf)a»  and  the  temperature  should  rise  another  degree. 
After  x  similar  reductions  the  volume  becomes  (HI)37'  and 
the  temperature  should  be  raised  x  degrees.  If  we  suppose 
({{l)x  —  T^,  and  if  we  take  the  logarithms  of  both,  we  find 

x  -  about  300°. 
If  we  suppose  (Hf)x  =  i>  we  find 

ar=80°; 
which  shows  that  air  compressed  one  half  rises  80°. 

All  this  is  subject  to  the  hypothesis  that  the  specific  heat 
of  air  does  not  change,  although  the  volume  diminishes. 
But  if,  for  the  reasons  hereafter  given  (pp.  86,  89),  we  re- 
gard the  specific  heat  of  air  compressed  one  half  as 
reduced  in  the  relation  of  700  to  616,  the  number  80°  must 
be  multiplied  by  |ff,  which  raises  it  to  90°. 


84  MOTIVE  POWER  OF  HEAT. 

that  they  be  taken  and  maintained  at  a  certain  in- 
variable temperature.  But  these  theories  furnish 
no  means  of  comparing  the  quantities  of  heat  liber- 
ated or  absorbed  by  elastic  fluids  which  change  in 
volume  at  different  temperatures.  Thus  we  are 
ignorant  what  relation  exists  between  the  heat  re- 
linquished by  a  litre  of  air  reduced  one  half,  the 
temperature  being  kept  at  zero,  and  the  heat  relin- 
quished by  the  same  litre  of  air  reduced  one  half, 
the  temperature  being  kept  at  100°.  The  knowl- 
edge of  this  relation  is  closely  connected  with  that 
of  the  specific  heat  of  gases  at  various  temperatures, 
and  to  some  other  data  that  Physics  as  yet  does  not 
supply. 

The  second  of  our  theorems  offers  us  a  means  of 
determining  according  to  what  law  the  specific 
heat  of  gases  varies  with  their  density. 

Let  us  suppose  that  the  operations  described  on 
p.  70,  instead  of  being  performed  with  two  bodies, 
A,  B,  of  temperatures  differing  indefinitely  small, 
were  carried  on  with  two  bodies  whose  tempera- 
tures differ  by  a  finite  quantity— one  degree,  for 
example.  In  a  complete  circle  of  operations  the 
body  A  furnishes  to  the  elastic  fluid  a  certain  quan- 
tity of  heat,  which  may  be  divided  into  two  por- 
tions :  (1)  That  which  is  necessary  to  maintain  the 
temperature  of  the  fluid  constant  during  dilata- 


MOTIVE  POWER  OF  HEAT.  85 

tion;  (2)  that  which  is  necessary  to  restore  the  tem- 
perature of  the  fluid  from  that  of  the  body  B  to 
that  of  the  body  A,  when,  after  having  brought 
back  this  fluid  to  its  primitive  volume,  we  place  it 
again  in  contact  with  the  body  A.  Let  us  call  the 
first  of  these  quantities  a  and  the  second  ~b.  The 
total  caloric  furnished  by  the  body  A  will  be  ex- 
pressed by  a  -\-  b. 

The  caloric  transmitted  by  the  fluid  to  the  body 
B  may  also  be  divided  into  two  parts :  one,  Z>',  due 
to  the  cooling  of  the  gas  by  the  body  B ;  the  other, 
a',  which  the  gas  abandons  as  a  result  of  its  re- 
duction of  volume.  The  sum  of  these  two  quanti- 
ties is  a'  -j-  V ')  it  should  be  equal  to  a  -j-  #,  for, 
after  a  complete  cycle  of  operations,  the  gas  is 
brought  back  exactly  to  its  primitive  state.  It  has 
been  obliged  to  give  up  all  the  caloric  which  has 
first  been  furnished  to  it.  We  have  then 

a+  b  =  a'  +  b'; 
or  rather, 

a  -  a'  =  V  -  I. 

Now,  according  to  the  theorem  given  on  page  81, 
the  quantities  a  and  a'  are  independent  of  the  den- 
sity of  the  gas,  provided  always  that  the  ponderable 
quantity  remains  the  same  and  that  the  variations 
of  volume  be  proportional  to  the  original  volume. 


86  MOTIVE  POWER  OF  HEAT. 

The  difference  a  —  a'  should  fulfil  the  same  condi- 
tions, and  consequently  also  the  difference  V  —  b, 
which  is  equal  to  it.  But  b'  is  the  caloric  neces- 
sary to  raise  the  gas  enclosed  in  abed  (Fig.  2)  one  de- 
gree ;  b'  is  the  caloric  surrendered  by  the  gas  when, 
enclosed  in  abcf,  it  is  cooled  one  degree.  These 
quantities  may  serve  as  a  measure  for  specific  heats. 
We  are  then  led  to  the  establishment  of  the  follow- 
ing proposition: 

The  client ge  in  the  specific  heat  of  a  gas  caused 
by  change  of  volume  depends  entirely  on  the  ratio 
between  the  original  volume  and  the  altered  volume. 
That  is,  the  difference  of  the  specific  heats  does  not 
depend  on  the  absolute  magnitude  of  the  volumes, 
but  only  on  their  ratio. 

This  proposition  might  also  be  differently  ex- 
pressed, thus: 

When  a  gas  increases  in  volume  in  geometrical 
progression,  its  specific  heat  increases  in  arith- 
metical progression. 

Thus,  a  being  the  specific  heat  of  air  taken  at  a 
given  density,  and  a  -\-  h  the  specific  heat  for  a 
density  one  half  less,  it  will  be,  for  a  density  equal 
to  one  quarter,  a  -f-  2h;  for  a  density  equal  to  one 
eighth,  a  -f-  37^ ;  and  so  on. 

The  specific  heats  are  here  taken  with  reference 
to  weight.  They  are  supposed  to  be  taken  at  an 


MOTIVE  POWER  OF  HEAT.  87 

invariable  volume,  but,  ;  s  we  shall  see,  they  would 
follow  the  same  law  if  they  were  taken  under  con- 
stant pressure. 

To  what  cause  is  the  difference  between  specific 
heats  at  constant  volume  and  at  constant  pressure 
really  due  ?  To  the  caloric  required  to  produce  in 
the  second  case  increase  of  volume.  Now,  accord- 
ing to  the  law  of  Mariotte,  increase  of  volume  of  a 
gas  should  be,  for  a  given  change  of  temperature, 
a  determined  fraction  of  the  original  volume,  a  . 
fraction  independent  of  pressure.  According  to 
the  theorem  expressed  on  page  76,  if  the  ratio  be- 
tween the  primitive  volume  and  the  altered  volume 
is  given,  that  determines  the  heat  necessary  to  pro- 
duce increase  of  volume.  It  depends  solely  on  this 
ratio  and  on  the  weight  of  the  gas.  We  must  then 
conclude  that : 

The  difference  between  specific  heat  at  constant 
pressure  and  specific  heat  at  constant  volume  is 
alivays  the  same,  whatever  may  be  the  density  of  the 
gas,  provided  the  weight  remains  the  same. 

These  specific  heats  both  increase  accordingly  as 
the  density  of  the  gas  diminishes,  but  their  differ- 
ence does  not  vary.* 

*MM.  Gay-Lussac  and  Welter  have  found  by  direct 
experiments,  cited  in  the  Mecanique  Celeste  and  in  the 
Annales  de  Chimie  et  de  Physique,  July,  1822,  p.  267,  that 


88  MOTIVE  POWEH  OF  HEAT. 

Since  the  difference  between  the  two  capacities 
for  heat  is  constant,  if  one  increases  in  arithmetical 
progression  the  other  should  follow  a  similar  pro- 
gression: thus  one  law  is  applicable  to  specific 
heats  at  constant  pressure. 

We  have  tacitly  assumed  the  increase  of  specific 
heat  with  that  of  volume.  This  increase  is  indi- 
cated by  the  experiments  of  MM.  Delaroche  and 
Berard:  in  fact  these  physicists  have  found  0.967 
for  the  specific  heat  of  air  under  the  pressure  of 

the  ratio  between  the  specific  heat  at  constant  pressure  and 
the  specific  heat  at  constant  volume  varies  very  little  with 
the  density  of  the  gas.  According  to  what  we  have  just 
seen,  the  difference  should  remain  constant,  and  not  the 
ratio.  As,  further,  the  specific  heat  of  gases  for  a  given 
weight  varies  very  little  with  the  density,  it  is  evident  that 
the  ratio  itself  experiences  but  slight  changes. 

The  ratio  between  the  specific  heat  of  atmospheric  air  at 
constant  pressure  and  at  constant  volume  is,  according 
to  MM.  Gay-Lussac  and  Welter,  1.3748,  a  number  almost 
constant  for  all  pressures,  and  even  for  all  temperatures. 
We  have  come,  through  other  considerations,  to  the  number 
^_116  =  1.44,  which  differs  from  the  former  ^,  and  we 
have  used  this  number  to  prepare  a  table  of  the  specific 
heats  of  gases  at  constant  volume.  So  we  need  not  regard 
this  table  as  very  exact,  any  more  than  the  table  given  on 
p.  89.  These  tables  are  mainly  intended  to  demonstrate 
the  laws  governing  specific  heats  of  aeriform  fluids. 


MOTIVE  POWEH  OF  HEAT. 


1  metre  of  mercury  (see  Memoire  already  cited), 
taking  for  the  unit  the  specific  heat  of  the  same 
weight  of  air  under  the  pressure  of  Om.7GO. 

According  to  the  law  that  specific  heats  follow 
with  relation  to  pressures,  it  is  only  necessary  to 
have  observed  them  in  two  particular  cases  to 
deduce  them  in  all  possible  cases  :  it  is  thus  that, 
making  use  of  the  experimental  result  of  MM. 
Delaroche  and  Berard  which  has  just  been  given, 
we  have  prepared  the  following  table  of  the  specific 
heat  of  air  under  different  pressures: 

SPECIFIC  HEAT  OF  Am. 


Pressure  in 

Specific  Heat, 
that  of  Air  under 

Pressure  in 

Specific  Heat, 
that  of  Air  under 

Atmospheres. 

Atmospheric  Pres- 
sure being  1. 

Atmospheres. 

Atmospheric  Pres- 
sure being  1. 

TTFUT 

1.840 

1 

1.000 

5  \~% 

1.756 

2 

0.916 

'ET>'?f 

1.672 

4 

0.832 

1.588 

8 

0.748 

w 

1.504 

16 

0.664 

S 

1.420 

32 

0.580 

TV 

1.336 

64 

0.496 

1.252 

128 

0.412 

1 

1.165 

256 

0.328 

1 

1.084 

512 

0.244 

1 

1.000 

1024 

0.160 

The  first  column  is,  as  we  see,  a  geometrical 
progression,  and  the  second  an  arithmetical  pro- 
gression. 


90  MOTIVE  POWER  OF  HEAT. 

We  have  carried  out  the  table  to  the  extremes 
of  compression  and  rarefaction.  It  may  be  be- 
lieved that  air  would  be  liquefied  before  acquiring 
a  density  1024  times  its  normal  density,  that  is, 
before  becoming  more  dense  than  water.  The 
specific  heat  would  become  zero  and  even  negative 
on  extending  the  table  beyond  the  last  term.  We 
think,  furthermore,  that  the  figures  of  the  second 
column  here  decrease  too  rapidly.  The  experi- 
ments which  serve  as  a  basis  for  our  calculation 
have  been  made  within  too  contracted  limits  for  us 
to  expect  great  exactness  in  the  figures  which  we 
have  obtained,  especially  in  the  outside  numbers. 

Since  we  know,  on  the  one  hand,  the  law  ac- 
cording to  which  heat  is  disengaged  in  the  com- 
pression of  gases,  and  on  the  other,  the  law  accord- 
ing to  which  specific  heat  varies  with  volume,  it 
will  be  easy  for  us  to  calculate  the  increase  of  tem- 
perature of  a  gas  that  has  been  compressed  with- 
out being  allowed  to  lose  heat.  In  fact,  the  com- 
pression may  be  considered  as  composed  of  two 
successive  operations :  (1)  compression  at  a  con- 
stant temperature  ;  (2)  restoration  of  the  caloric 
emitted.  The  temperature  will  rise  through  the 
second  operation  in  inverse  ratio  with  the  specific 
heat  acquired  by  the  gas  after  the  reduction  of 
volume, — specific  heat  that  we  are  able  to  calculate 


MOTIVE  POWER  OF  HEAT.  91 

by  means  of  the  law  demonstrated  above.  The 
heat  set  free  by  compression,  according  to  the 
theorem  of  page  81,  ought  to  be  represented  by  an 
expression  of  the  form 

s  =  A  +  B  log  v, 

s  being  this  heat,  v  the  volume  of  the  gas  after 
compression,  A  and  B  arbitrary  constants  depen- 
dent on  the  primitive  volume  of  the  gas,  on  its 
pressure,  and  on  the  units  chosen. 

The  specific  heat  varying  with  the  volume  ac- 
cording to  the  law  just  demonstrated,  should  be 
represented  by  an  expression  of  the  form 

z  =  A'  +  B'  log  v, 

A'  and  B'  being  the  different  arbitrary  constants 
of  A  and  B. 

The  increase  of  temperature  acquired  by  the 
gas,  as  the  effect  of  compression,  is  proportional  to 

the  ratio   -  or  to  the  relation     .,,„,,        •     It 
z  A'  +  B'  log  v 

can  be  represented  by  this  ratio  itself;  thus,  calling 
it  t,  we  shall  have 

—  A   +B  logv 
~  A'  +  H'Iogv 

If  the  original  volume  of  the  gas  is  1,  and  the 
original  temperature  zero,  we  shall  have  at  the 


92  MOTIVE  POWEH  OF  HEAT. 

same  time  t  =  0,  log  v  =  0,  whence  A .  =  0  ;  ^  will 
then  express  not  only  the  increase  of  temperature, 
but  the  temperature  itself  above  the  thermometric 
zero. 

We  need  not  consider  the  formula  that  we  have 
just  given  as  applicable  to  very  great  changes  in 
the  volume  of  gases.  We  have  regarded  the  ele- 
vation of  temperature  as  being  in  inverse  ratio  to 
the  specific  heat;  which  tacitly  supposes  the  specific 
heat  to  be  constant  at  all  temperatures.  Great 
changes  of  volume  lead  to  great  changes  of  tem- 
perature in  the  gas,  and  nothing  proves  the  con- 
stancy of  specific  heat  at  different  temperatures, 
especially  at  temperatures  widely  separated.  This 
constancy  is  only  an  hypothesis  admitted  for  gases 
by  analogy,  to  a  certain  extent  verified  for  solid 
bodies  and  liquids  throughout  a  part  of  the  ther- 
mometric scale,  but  of  which  the  experiments  of 
MM.  Dulong  and  Petit  have  shown  the  inaccuracy 
when  it  is  desirable  to  extend  it  to  temperatures 
far  above  100°.* 

According  to  a  law  of  MM.  Clement  and  De- 
sormes,  a  law  established  by  direct  experiment,  the 
vapor  of  water,  under  whatever  pressure  it  may 
be  formed,  contains  always,  at  equal  weights,  the 

*  Note  C,  Appendix  B. 


MOTIVE  POWER  OF  HEAT.  93 

same  quantity  of  heat;  which  leads  to  the  assertion 
that  steam,  compressed  or  expanded  mechanically 
without  loss  of  heat,  will  always  be  found  in  a 
saturated  state  if  it  was  so  produced  in  the  first 
place.  The  vapor  of  water  so  made  may  then  be 
regarded  as  a  permanent  gas,  and  should  observe 
all  the  laws  of  one.  Consequently  the  formula 

A   +  B  log  v 
~  A'  +  B'  log  v 

should  be  applicable  to  it,  and  be  found  to  accord 
with  the  table  of  tensions  derived  from  the  direct 
experiments  of  M.  Dalton. 

We  may  be  assured,  in  fact,  that  our  formula, 
with  a  convenient  determination  of  arbitrary  con- 
stants, represents  very  closely  the  results  of  experi- 
ment. The  slight  irregularities  which  we  find 
therein  do  not  exceed  what  we  might  reasonably 
attribute  to  errors  of  observation.* 

We  will  return,  however,  to  our  principal  sub- 
ject, from  which  we  have  wandered  too  far — the 
motive  power  of  heat. 

We  have  shown  that  the  quantity  of  motive 
power  developed  by  the  transfer  of  caloric  from 
one  body  to  another  depends  essentially  upon  the 
temperature  of  the  two  bodies,  but  we  have  not 

*  Note  D,  Appendix  B. 


94  MOTIVE  POWER  OF  HEAT. 

shown  the  relation  between  these  temperatures  and 
the  quantities  of  motive  power  produced.  It  would 
at  first  seem  natural  enough  to  suppose  that  for 
equal  differences  of  temperature  the  quantities  of 
motive  power  produced  are  equal ;  that  is,  for  ex- 
ample, the  passage  of  a  given  quantity  of  caloric 
from  a  body,  A,  maintained  at  100°,  to  a  body,  B, 
maintained  at  50°,  should  give  rise  to  a  quantity  of 
motive  power  equal  to  that  which  would  be  devel- 
oped by  the  transfer  of  the  same  caloric  from  a 
body,  B,  at  50°,  to  a  body,  C,  at  zero.  Such  a  law 
would  doubtless  be  very  remarkable,  but  we  do  not 
see  sufficient  reason  for  admitting  it  a  priori.  We 
will  investigate  its  reality  by  exact  reasoning. 

Let  us  imagine  that  the  operations  described  on 
p.  70  be  conducted  successively  on  two  quantities 
of  atmospheric  air  equal  in  weight  and  volume, 
but  taken  at  different  temperatures.  Let  us  sup- 
pose, further,  the  differences  of  temperature  be- 
tween the  bodies  A  and  B  equal,  so  these  bodies 
would  have  for  example,  in  one  of  these  cases,  the 
temperatures  100°  and  100°  —  h  (h  being  indefi- 
nitely small),  and  in  the  other  1°  and  1°  —  h.  The 
quantity  of  motive  power  produced  is,  in  each  case, 
the  difference  between  that  which  the  gas  supplies 
by  its  dilatation  and  that  which  must  be  expended 
to  restore  its  primitive  volume.  Now  this  differ- 


MOTIVE  POWER  OF  HEAT.  95 

ence  is  the  same  in  both  cases,  as  any  one  can 
prove  by  simple  reasoning,  which  it  seems  un- 
necessary to  give  here  in  detail ;  hence  the  motive 
power  produced  is  the  same. 

Let  us  now  compare  the  quantities  of  heat  em- 
ployed in  the  two  cases.  In  the  first,  the  quantity 
of  heat  employed  is  that  which  the  body  A  fur- 
nishes to  the  air  to  maintain  it  at  the  temperature 
of  100°  during  its  expansion.  In  the  second,  it  is 
the  quantity  of  heat  which  this  same  body  should 
furnish  to  it,  to  keep  its  temperature  at  one  degree 
during  an  exactly  similar  change  of  volume.  If 
these  two  quantities  of  heat  were  equal,  there 
would  evidently  result  the  law  that  we  have  already 
assumed.  But  nothing  proves  that  it  is  so,  and  we 
shall  find  that  these  quantities  are  not  equal. 

The  air  that  we  shall  first  consider  as  occupying 
the  space  abed  (Fig.  2),  and  having  1  degree  of 
temperature,  can  be  made  to  occupy  the  space  abef, 
and  to  acquire  the  temperature  of  100  degrees  by 
two  different  means: 

(1)  We  may  heat  it  without  changing  its  vol- 
ume,   then   expand   it,    keeping  its   temperature 
constant. 

(2)  We  may  begin  by  expanding  it,  maintaining 
the  temperature  constant,  then  heat  it,  when  it 
has  acquired  its  greater  volume. 


96  MOTIVE  POWER  OF  HEAT. 

Let  a  and  1}  be  the  quantities  of  heat  employed 
successively  in  the  first  of  the  two  operations,  and 
let  V  and  a'  be  the  quantities  of  heat  employed 
successively  in  the  second.  As  the  final  result  of 
these  two  operations  is  the  same,  the  quantities  of 
heat  employed  in  both  should  be  equal.  We  have 

then 

a  +  b  =  a'  +  V, 

whence 

a'  -  a  =  b  -b'. 

a'  is  the  quantity  of  heat  required  to  cause  the 
gas  to  rise  from  1°  to  100°  when  it  occupies  the 
space  abef. 

a  is  the  quantity  of  heat  required  to  cause  the 
gas  to  rise  from  1°  to  100°  when  it  occupies  the 
space  abed. 

The  density  of  the  air  is  less  in  the  first  than  in 
the  second  case,  and  according  to  the  experiments 
of  MM.  Delaroche  and  Berard,  already  cited  on 
page  87,  its  capacity  for  heat  should  be  a  little 
greater. 

The  quantity  a'  being  found  to  be  greater  than 
the  quantity  a,  b  should  be  greater  than  b'.  Con- 
sequently, generalizing  the  proposition,  we  should 
say: 

The  quantity  of  heat  due  to  the  change  of  volume 
of  a  gas  is  greater  as  the  temperature  is  higher. 


MOTIVE  POWER  OF  HEAT.  97 

Thus,  for  example,  more  caloric  is  necessary  to 
maintain  at  100°  the  temperature  of  a  certain 
quantity  of  air  the  volume  of  which  is  doubled, 
than  to  maintain  at  1°  the  temperature  of  this 
same  air  during  a  dilatation  exactly  equal. 

These  unequal  quantities  of  heat  would  produce, 
however,  as  we  have  seen,  equal  quantities  of 
motive  power  for  equal  fall  of  caloric  taken  at  dif- 
ferent heights  on  the  thermometric  scale;  whence 
we  draw  the  following  conclusion : 

The  fall  of  caloric  produces  more  motive  power  at 
inferior  than  at  superior  temperatures. 

Thus  a  given  quantity  of  heat  will  develop  more 
motive  power  in  passing  from  a  body  kept  at  1 
degree  to  another  maintained  at  zero,  than  if  these 
two  bodies  were  at  the  temperature  of  101°  and 
100°. 

The  difference,  however,  should  be  very  slight. 
It  would  be  nothing  if  the  capacity  of  the  air  for 
heat  remained  constant,  in  spite  of  changes  of 
density.  According  to  the  experiments  of  MM. 
Delaroche  and  Berard,  this  capacity  varies  little — 
so  little  even,  that  the  differences  noticed  might 
strictly  have  been  attributed  to  errors  of  observa- 
tion or  to  some  circumstances  of  which  we  have 
failed  to  take  account. 

We  are   not   prepared   to  determine   precisely, 


98  MOTIVE  POWER  OF  HEAT. 

with  no  more  experimental  data  than  we  now  pos- 
sess, the  law  according  to  which  the  motive  power 
of  heat  varies  at  different  points  on  the  ther mo- 
metric  scale.  This  law  is  intimately  connected 
with  that  of  the  variations  of  the  specific  heat  of 
gases  at  different  temperatures — a  law  which  ex- 
periment has  not  yet  made  known  to  us  with  suffi- 
cient exactness.* 

We  will  endeavor  now  to  estimate  exactly  the 
motive  power  of  heat,  and  in  order  to  verify  our 
fundamental  proposition,  in  order  to  determine 
whether  the  agent  used  to  realize  the  motive  power 
is  really  unimportant  relatively  to  the  quantity  of 
this  power,  we  will  select  several  of  them  succes- 
sively: atmospheric  air,  vapor  of  water,  vapor  of 
alcohol. 

Let  us  suppose  that  we  take  first  atmospheric 
air.  The  operation  will  proceed  according  to  the 
method  indicated  on  page  70.  We  will  make  the 
following  hypotheses :  The  air  is  taken  under 
atmospheric  pressure.  The  temperature  of  the 
body  A  is  y^r  °^  a  degree  above  zero,  that  of  the 
body  B  is  zero.  The  difference  is,  as  we  see,  very 
slight — a  necessary  condition  here. 

The  increase  of  volume  given  to  the  air  in  our 

*  Note  E,  Appendix  B. 


MOTIVE  POWER  OF  HEAT.  99 

operation  will  be  TI7  +  ^T  of  the  primitive  vol- 
ume ;  this  is  a  very  slight  increase,  absolutely 
speaking,  but  great  relatively  to  the  difference  of 
temperature  between  the  bodies  A  and  B. 

The  motive  power  developed  by  the  whole  of 
the  two  operations  described  (page  70)  will  be  very 
nearly  proportional  to  the  increase  of  volume  and 
to  the  difference  between  the  two  pressures  exer- 
cised by  the  air,  when  it  is  found  at  the  tempera- 
tures 0°.001  and  zero. 

This  difference  is,  according  to  the  law  of  M. 
Gay-Lussac,  ^Wo^o  °^  *ne  elastic  force  of  the  gas, 
or  very  nearly  ^g^VinF  °f  *ne  atmospheric  pressure. 

The  atmospheric  pressure  balances  at  10.40 
metres  head  of  water  ;  wfop$  °f  this  pressure 
equals  -g-^VoFo  X  10m.40  of  head  of  water. 

As  to  the  increase  of  volume,  it  is,  by  supposi- 
tion, yj-g-  +  ^-T  of  the  original  volume,  that  is,  of 
the  volume  occupied  by  one  kilogram  of  air  at 
zero,  a  volume  equal  to  Omc.77,  allowing  for  the 
specific  weight  of  the  air.  So  then  the  product, 


will  express  the  motive  power  developed.  This 
]  ower  is  estimated  here  in  cubic  metres  of  water 
raised  one  metre, 


100  MOTIVE  POWER  OF  HEAT. 

If  we  carry  out  the  indicated  multiplications,  we 
find  the  value  of  the  product  to  be  0.000000372. 

Let  us  endeavor  now  to  estimate  the  quantity  of 
heat  employed  to  give  this  result  ;  that  is,  the 
quantity  of  heat  passed  from  the  body  A  to  the 
body  B. 

The  body  A  furnishes  : 

(1)  The  heat  required  to  carry  the  temperature 
of  one  kilogram  of  air  from  zero  to  0°.001; 

(2)  The  quantity  necessary  to  maintain  at  this 
temperature  the  temperature  of  the  air  when  it 
experiences  a  dilatation  of 


TTTT  ~T 


The  first  of  these  quantities  of  heat  being  very 
small  in  comparison  with  the  second,  we  may  dis- 
regard it.  The  second  is,  according  to  the  rea- 
soning on  page  74,  equal  to  that  which  would  be 
necessary  to  increase  one  degree  the  temperature 
of  one  kilogram  of  air  subjected  to  atmospheric 
pressure. 

According  to  the  experiments  of  MM.  Delaroche 
and  Berard  on  the  specific  heat  of  gases,  that  of 
air  is,  for  equal  weights,  0.267  that  of  water.  If, 
then,  we  take  for  the  unit  of  heat  the  quantity 
necessary  to  raise  1  kilogram  of  water  1  degree, 


MOTIVE  POWER  OF  SEAT.  101 

that  which  will  be  required  to  raise  1  kilogram  of 
air  1  degree  would  have  for  its  value  0.267.  Thus 
the  quantity  of  heat  furnished  by  the  body  A  is 

0.267  units. 

This  is  the  heat  capable  of  producing  0.000000372 
units  of  motive  power  by  its  fall  from  0°.001  to 
zero. 

For  a  fall  a  thousand  times  greater,  for  a  fall  of 
one  degree,  the  motive  power  will  be  very  nearly  a 
thousand  times  the  former,  or 

0.000372. 

If,  now,  instead  of  0.267  units  of  heat  we  employ 
1000  units,  the  motive  power  produced  will  be 
expressed  by  the  proportion 

0.267          1000       ,  372 

-,    whence    x  =  ^--  =  1.395. 


0.000372          x   '  267 

Thus  1000  units  of  heat  passing  from  a  body 
maintained  at  the  temperature  of  1  degree  to 
another  body  maintained  at  zero  would  produce,  in 
acting  upon  the  air, 

1.395  units  of  motive  power. 

We  will  now  compare  this  result  with  that  fur- 
nished by  the  action  of  heat  on  the  vapor  of  water, 


102  MOTIVE  POWER  OF  HEAT. 

Let  us  suppose  one  kilogram  of  liquid  water  en- 
closed in  the  cylindrical  vessel  abed  (Fig.  4),  be- 
tween the  bottom  ab  and  the  piston 
cd.  Let  us  suppose,  also,  the  two 
bodies  A,  B  maintained  each  at  a 
constant  temperature,  that  of  A  being 
a  very  little  above  that  of  B.  Let  us 
imagine  now  the  following  operations : 
(1)  Contact  of  the  water  with  the 
body  A,  movement  of  the  piston  from 
the  position  cd  to  the  position  ef,  for- 
mation of  steam  at  the  temperature 
of  the  body  A  to  fill  the  vacuum  pro- 
duced by  the  extension  of  volume.  We  will  sup- 
pose the  space  abef  large  enough  to  contain  all  the 
water  in  a  state  of  vapor. 

(2)  Removal  of  the  body  A,  contact  of  the  vapor 
with  the  body  B,  precipitation  of  a  part  of  this 
vapor,  diminution  of  its  elastic  force,  return  of 
the  piston  from  ef  to  ab,  liquefaction  of  the  rest  of 
the  vapor  through  the  effect  of  the  pressure  com- 
bined with  the  contact  of  the  body  B. 

(3)  Removal  of  the  body  B,  fresh   contact  of 
the  water  with  the  body  A,  return  of  the  water 
to  the  temperature  of  this  body,  renewal  of  the 
former  period,  and  so  on. 

The  quantity  of  motive  power  developed  in  a 


MOTIVE  POWER  OF  HEAT.  103 

complete  cycle  of  operations  is  measured  by  the 
product  of  the  volume  of  the  vapor  multiplied  by 
the  difference  between  the  tensions  that  it  pos- 
sesses at  the  temperature  of  the  body  A  and  at 
that  of  the  body  B.  As  to  the  heat  employed, 
that  is  to  say,  transported  from  the  body  A  to  the 
body  B,  it  is  evidently  that  which  was  necessary 
to  turn  the  water  into  vapor,  disregarding  always 
the  small  quantity  required  to  restore  the  tempera- 
ture of  the  liquid  water  from  that  of  B  to  that 
of  A. 

Suppose  the  temperature  of  the  body  A  100  de- 
grees, and  that  of  the  body  .Z?  99  degrees:  the 
difference  of  the  tensions  will  be,  according  to  the 
table  of  M.  Dalton,  26  millimetres  of  mercury  or 
Om.36  head  of  water. 

The  volume  of  the  vapor  is  1700  times  that  of 
the  water.  If  we  operate  on  one  kilogram,  that 
will  be  1700  litres,  or  lrac.700. 

Thus  the  value  of  the  motive  power  developed 
is  the  product 

1.700  X  0.36  =0.611  units, 

of  the  kind  of  which  we  have  previously  made  use. 

The  quantity  of  heat  employed  is  the  quantity 

required  to  turn  into  vapor  water  already  heated  to 

100°.    This  quantity  is  found  by  experiment.    "We 


104  MOTIVE  POWER  OF  HEAT. 

have  found  it  equal  to  550°,  or,   to  speak  more 
exactly,  to  550  of  our  units  of  heat. 

Thus  0.611  units  of  motive  power  result  from 
the  employment  of  550  units  of  heat.  The  quan- 
tity of  motive  power  resulting  from  1000  units  of 
heat  will  be  given  by  the  proportion 

550         1000  611 

whence    x  =  —--  =  1.112. 


0.611         x  550 

Thus  1000  units  of  heat  transported  from  one 
body  kept  at  100  degrees  to  another  kept  at  99 
degrees  will  produce,  acting  upon  vapor  of  water, 
1.112  units  of  motive  power. 

The  number  1.112  differs  by  about  J  from  the 
number  1.395  previously  found  for  the  value  of  the 
motive  power  developed  by  1000  units  of  heat  acting 
upon  the  air ;  but  it  should  be  observed  that  in  this 
case  the  temperatures  of  the  bodies  A  and  B  were 
1  degree  and  zero,  while  here  they  are  100  degrees 
and  99  degrees.  The  difference  is  much  the  same  ; 
but  it  is  not  found  at  the  same  height  in  the  ther- 
mometric  scale.  To  make  an  exact  comparison,  it 
would  have  been  necessary  to  estimate  the  motive 
power  developed  by  the  steam  formed  at  1  degree 
and  condensed  at  zero.  It  would  also  have  been 
necessary  to  know  the  quantity  of  heat  contained 
in  the  steam  formed  at  one  degree. 


MOTIVE  POWER  OF  HEAT.  105 

The  law  of  MM.  Clement  and  Desormes  re- 
ferred to  on  page  92  gives  ns  this  datum.  The 
constituent  heat  of  vapor  of  water  being  always  the 
same  at  any  temperature  at  which  vaporization 
takes  place,  if  550  degrees  of  heat  are  required  to 
vaporize  water  already  brought  up  to  100  degrees, 
550  -f- 100  or  650  will  be  required  to  vaporize  the 
same  weight  of  water  taken  at  zero. 

Making  use  of  this  datum  and  reasoning  exactly 
as  we  did  for  water  at  100  degrees,  we  find,  as  is 
easily  seen, 

1.290 

for  the  motive  power  developed  by  1000  units  of 
heat  acting  upon  the  vapor  of  water  between  one 
degree  and  zero.  This  number  approximates  more 
closely  than  the  first  to 

1.395. 

It  differs  from  it  only  T*j,  an  error  which  does  not 
exceed  probable  limits,  considering  the  great  num- 
ber of  data  of  different  sorts  of  which  we  have 
been  obliged  to  make  use  in  order  to  arrive  at  this 
approximation.  Thus  is  our  fundamental  law  veri- 
fied in  a  special  case.* 

*  We  find  (Annales  de  Chimie  et  de  Physique,  July,  1818, 
p.  294)  in  a  memoir  of  M.  Petit  an  estimate  of  the  motive 
power  of  heat  applied  to  air  and  to  vapor  of  water.  This 


106  MOTIVE  POWER  OF  HEAT. 

We  will  examine  another  case  in  which  vapor  of 
alcohol  is  acted  upon  by  heat.  The  reasoning  is 
precisely  the  same  as  for  the  vapor  of  water.  The 
data  alone  are  changed.  Pure  alcohol  boils  under 
ordinary  pressure  at  78°.7  Centigrade.  One  kilo- 
gram absorbs,  according  to  MM.  Delaroche  and 
Berard,  207  units  of  heat  in  undergoing  transfor- 
mation into  vapor  at  this  same  temperature,  78°.7. 

The  tension  of  the  vapor  of  alcohol  at  one  de- 
gree below  the  boiling-point  is  found  to  be  dimin- 
ished -gig-.  It  is  2^  less  than  the  atmospheric 
pressure ;  at  least,  this  is  the  result  of  the  experi- 
ment of  M.  Betancour  reported  in  the  second  part 
of  V Architecture  hydraulique  of  M.  Prony,  pp. 
180,  195.* 

If  we  use  these  data,  we  find  that,  in  acting  upon 
one  kilogram  of  alcohol  at  the  temperatures  of 
78°.  7  and  77°.  7,  the  motive  power  developed  will 
be  0.251  units. 

This  results  from  the  employment  of  207  units 
of  heat.  For  1000  units  the  proportion  must  be 

207        1000 


0.254 


whence    x  =  1.230. 


estimate  leads  us  to  attribute  a  great  advantage  to  atmos- 
pheric air,  but  it  is  derived  by  a  method  of  considering  the 
action  of  heat  which  is  quite  imperfect. 
*  Note  F,  Appendix  B. 


MOTIVE  POWER  OF  HEAT.  107 

This  number  is  a  little  more  than  the  1.112  re- 
sulting from  the  use  of  the  vapor  of  water  at  tb.e 
temperatures  100°  and  99°;  but  if  we  suppose  the 
vapor  of  water  used  at  the  temperatures  78°  and 
77°,  we  find,  according  to  the  law  of  MM.  Clement 
and  Desorme,  1.212  for  the  motive  power  due  to 
1000  units  of  heat.  This  latter  number  ap- 
proaches, as  we  see,  very  nearly  to  1.230.  There 
is  a  difference  of  only  ^. 

We  should  have  liked  to  be  able  to  make  other 
approximations  of  this  sort — to  be  able  to  calculate, 
for  example,  the  motive  power  developed  by  the 
action  of  heat  on  solids  and  liquids,  by  the  conge- 
lation of  water,  and  so  on;  but  Physics  as  yet  re- 
fuses us  the  necessary  data0* 

The  fundamental  law  that  we  propose  to  confirm 
seems  to  us  to  require,  however,  in  order  to  be 
placed  beyond  doubt,  new  verifications.  It  is  based 
upon  the  theory  of  heat  as  it  is  understood  to-day, 
and  it  should  be  said  that  this  foundation  does  not 
appear  to  be  of  unquestionable  solidity.  New  ex- 
periments alone  can  decide  the  question.  Mean- 
while we  can  apply  the  theoretical  ideas  expressed 

*  Those  that  we  need  are  the  expansive  force  acquired 
by  solids  and  liquids  by  a  given  increase  of  temperature, 
and  the  quantity  of  heat  absorbed  or  relinquished  in  the 
changes  of  volume  of  these  bodies. 


108  MOTIVE  POWER  OF  HEAT. 

above,  regarding  them  as  exact,  to  the  examination 
of  the  different  methods  proposed  up  tc  date,  for 
the  realization  of  the  motive  power  of  heat. 

It  has  sometimes  been  proposed  to  develop  mo- 
tive power  by  the  action  of  heat  on  solid  bodies. 
The  mode  of  procedure  which  naturally  first  occurs 
to  the  mind  is  to  fasten  immovably  a  solid  body — 
a  metallic  bar,  for  example — by  one  of  its  extremi- 
ties ;  to  attach  the  other  extremity  to  a  movable 
part  of  the  machine;  then,  by  successive  heating 
and  cooling,  to  cause  the  length  of  the  bar  to  vary, 
and  so  to  produce  motion.  Let  us  try  to  decide 
whether  this  method  of  developing  motive  power 
can  be  advantageous.  We  have  shown  that  the 
condition  of  the  most  effective  employment  of  heat 
in  the  production  of  motion  is,  that  all  changes 
of  temperature  occurring  in  the  bodies  should  be 
due  to  changes  of  volume.  The  nearer  we  come 
to  fulfilling  this  condition  the  more  fully  will  the 
heat  be  utilized.  Now,  working  in  the  manner 
just  described,  we  are  very  far  from  fulfilling  this 
condition  :  change  of  tempeiYiture  is  not  due  here 
to  change  of  volume  ;  all  the  changes  are  due  to 
contact  of  bodies  differently  heated — to  the  con- 
tact of  the  metallic  bar,  either  with  the  body 
charged  with  furnishing  heat  to  it,  or  with  the 
body  charged  with  carrying  it  off. 


MOTIVE  POWER  OF  HEAT. 


The  only  means  of  fulfilling  the  prescribed  con- 
dition would  be  to  act  upon  the  solid  body  exactly 
as  we  did  on  the  air  in  the  operations  described  on 
page  92.  But  for  this  we  must  be  able  to  pro- 
duce, by  a  single  change  of  volume  of  the  solid 
body,  considerable  changes  of  temperature,  that  is, 
if  we  should  want  to  utilize  considerable  falls  of 
caloric.  Now  this  appears  impracticable.  In 
short,  many  considerations  lead  to  the  conclusion 
that  the  changes  produced  in  the  temperature  of 
solid  or  liquid  bodies  through  the  effect  of  com- 
pression and  rarefaction  would  be  but  slight. 

(1)  We  often  observe  in  machines  (particularly 
in  steam-engines)  solid  pieces  which  endure  con- 
siderable   strain    in    one    way    or    another,   and 
although  these  efforts  may  be  sometimes  as  great 
as  the  nature  of  the  substances  employed  permits, 
the  variations  of  temperature  are  scarcely  per- 
ceptible. 

(2)  In  the  action  of  striking  medals,  in  that  of  the 
rolling-mill,  of  the  draw-plate,  the  metals  undergo 
the  greatest  compression  to  which  we  can  submit 
them,  employing  the  hardest  and  strongest  tools. 
Nevertheless  the  elevation  of  temperature  is  not 
great.     If  it  were,  the  pieces  of  steel  used  in  these 
operations  would  soon  lose  their  temper. 

(3)  We  know  that  it  would  be  necessary  to  exert 


110  MOTIVE  POWER  OF  HEAT. 

on  solids  and  liquids  a  very  great  strain  in  order  to 
produce  in  them  a  reduction  of  volume  comparable 
to  that  which  they  experience  in  cooling  (cooling 
from  100°  to  zero,  for  example).  Now  the  cooling 
requires  a  greater  abstraction  of  caloric  than  would 
simple  reduction  of  volume.  If  this  reduction 
were  produced  by  mechanical  means,  the  heat  set 
free  would  not  then  be  able  to  make  the  tempera- 
ture of  the  body  vary  as  many  degrees  as  the  cool- 
ing makes  it  vary.  It  would,  however,  necessitate 
the  employment  of  a  force  undoubtedly  very  con- 
siderable. 

Since  solid  bodies  are  susceptible  of  little  change 
of  temperature  through  changes  of  volume,  and 
since  the  condition  of  the  most  .effective  employ- 
ment of  heat  for  the  development  of  motive  power 
is  precisely  that  all  change  of  temperature  should  be 
due  to  a  change  of  volume,  solid  bodies  appear  but 

111  fitted  to  realize  this  power. 

The  same  remarks  apply  to  liquids.  The  same 
reasons  may  be  given  for  rejecting  them.* 

We  are  not  speaking  now  of  practical  difficulties. 

*  The  recent  experiments  of  M.  Oerstedt  on  the  com- 
pressibility of  water  have  shown  that,  for  a  pressure  of 
five  atmospheres,  the  temperature  of  this  liquid  exhibits 
no  appreciable  change.  (See  Annales  de  Ohimie  et  de 
Physique,  Feb.  1823,  p.  192.) 


MOTIVE  POWER  OF  HEAT.  Ill 

They  will  be  numberless.  The  motion  produced 
by  the  dilatation  and  compression  of  solid  or  liquid 
bodies  would  only  be  very  slight.  In  order  to  give 
them  sufficient  amplitude  we  should  be  forced  to 
make  use  of  complicated  mechanisms.  It  would 
be  necessary  to  employ  materials  of  the  greatest 
strength  to  transmit  enormous  pressure ;  finally, 
the  successive  operations  would  be  executed  very 
slowly  compared  to  those  of  the  ordinary  steam- 
engine,  so  that  apparatus  of  large  dimensions  and 
heavy  cost  would  produce  but  very  ordinary  re- 
sults. 

The  elastic  fluids,  gases  or  vapors,  are  the  means 
really  adapted  to  the  development  of  the  motive 
power  of  heat.  They  combine  all  the  conditions 
necessary  to  fulfil  this  office.  They  are  easy  to 
compress ;  they  can  be  almost  infinitely  expanded ; 
variations  of  volume  occasion  in  them  great 
changes  of  temperature;  and,  lastly,  they  are  very 
mobile,  easy  to  heat  and  to  cool,  easy  to  transport 
from  one  place  to  another,  which  enables  them  to 
produce  rapidly  the  desired  effects.  We  can  easily 
conceive  a  multitude  of  machines  fitted  to  develop 
the  motive  power  of  heat  through  the  use  of 
elastic  fluids  ;  but  in  whatever  way  we  look  at  it, 
we  should  not  lose  sight  of  the  following  prin- 
ciples: 


112  MOTIVE  POWER  OF  HEAT. 

(1)  The  temperature  of  the  fluid  should  be  made 
as  high  as  possible,  in  order  to  obtain  a  great  fall 
of  caloric,,  and  consequently  a  large  production  of 
motive  power. 

(2)  For  the  same  reason  the  cooling  should  be 
carried  as  far  as  possible. 

(3)  It  should  be  so  arranged  that  the  passage 
of  the  elastic  fluid  from  the  highest  to  the  lowest 
temperature  should  be  due  to  increase  of  volume; 
that  is,  it  should  be  so  arranged  that  the  cooling  of 
the  gas  should  occur  spontaneously  as  the  effect  of 
rarefaction.      The  limits  of  the  temperature  to 
which  it  is  possible  to  bring  the  fluid  primarily,, 
are  simply  the  limits  of  the  temperature  obtainable 
by  combustion  ;  they  are  very  high. 

The  limits  of  cooling  are  found  in  the  tempera- 
ture of  the  coldest  body  of  which  we  can  easily  and 
freely  make  use ;  this  body  is  usually  the  water  of 
the  locality. 

As  to  the  third  condition,  it  involves  difficulties 
in  the  realization  of  the  motive  power  of  heat 
when  the  attempt  is  made  to  take  advantage  of 
great  differences  of  temperature,  to  utilize  great 
falls  of  heat.  In  short,  it  is  necessary  then  that 
the  gas,  by  reason  of  its  rarefaction,  should  pass 
from  a  very  high  temperature  to  a  very  low  one, 
which  requires  a  great  change  of  volume  and  of 


MOTIVE  POWER  OF  HEAT.  113 

density,  which  requires  also  that  the  gas  be  first 
taken  under  a  very  heavy  pressure,  or  that  it 
acquire  by  its  dilatation  an  enormous  volume — 
conditions  both  difficult  to  fulfil.  The  first  neces- 
sitates the  employment  of  very  strong  vessels  to 
contain  the  gas  at  a  very  high  temperature  and 
under  very  heavy  pressure.  The  second  necessi- 
tates the  use  of  vessels  of  large  dimensions.  These 
are,  in  a  word,  the  principal  obstacles  which  pre- 
vent the  utilization  in  steam-engines  of  a  great 
part  of  the  motive  power  of  the  heat.  We  are 
obliged  to  limit  ourselves  to  the  use  of  a  slight  fall 
of  caloric,  while  the  combustion  of  the  coal  fur- 
nishes the  means  of  procuring  a  very  great  one. 

It  is  seldom  that  in  steam-engines  the  elastic 
fluid  is  produced  under  a  higher  pressure  than  six 
atmospheres — a  pressure  corresponding  to  about 
160°  Centigrade,  and  it  is  seldom  that  condensa- 
tion takes  place  at  a  temperature  much  under  40°. 
The  fall  of  caloric  from  160°  to  40°  is  120°,  while 
by  combustion  we  can  procure  a  fall  of  1000°  to 
2000°. 

In  order  to  comprehend  this  more  clearly,  let  us 
recall  what  we  have  termed  the  fall  of  caloric. 
This  is  the  passage  of  the  heat  from  one  body,  A, 
having  an  elevated  temperature,  to  another,  B, 
where  it  is  lower.  We  say  that  the  fall  of  the 


114  MOTIVE  POWER  OF  HEAT. 

caloric  is  100°  or  1000°  when  the  difference  of 
temperature  between  the  bodies  A  and  B  is  100° 
or  1000°. 

In  a  steam-engine  which  works  under  a  pressure 
of  six  atmospheres  the  temperature  of  the  boiler  is 
160°.  This  is  the  body  A.  It  is  kept,  by  contact 
with  the  furnace,  at  the  constant  temperature  of 
160°,  and  continually  furnishes  the  heat  necessary 
for  the  formation  of  steam.  The  condenser  is  the 
body  B.  By  means  of  a  current  of  cold  water  it 
is  kept  at  a  nearly  constant  temperature  of  40°.  It 
absorbs  continually  the  caloric  brought  from  the 
body  A  by  the  steam.  The  difference  of  tempera- 
ture between  these  two  bodies  is  160°  -  40°,  or  120°. 
Hence  we  say  that  the  fall  of  caloric  is  here  120°. 

Coal  being  capable  of  producing,  by  its  combus- 
tion, a  temperature  higher  than  1000°,  and  the 
cold  water,  which  is  generally  used  in  our  climate, 
being  at  about  10°,  we  can  easily  procure  a  fall  of 
caloric  of  1000°,  and  of  this  only  120°  are  utilized 
by  steam-engines.  Even  these  120°  are  not  wholly 
utilized.  There  is  always  considerable  loss  due 
to  useless  re-establishments  of  equilibrium  in  the 
caloric. 

It  is  easy  to  see  the  advantages  possessed  by 
high-pressure  machines  over  those  of  lower  pres- 
sure. This  superiority  lies  essentially  in  the  power 


MOTIVE  POWER  OF  HEAT. 


115 


of  utilizing  a  greater  fall  of  caloric.  The  steam 
produced  under  a  higher  pressure  is  found  also 
at  a  higher  temperature,  and  as,  further,  the 
temperature  of  condensation  remains  always  about 
the  same,  it  is  evident  that  the  fall  of  caloric  is 
more  considerable.  But  to  obtain  from  high-pres- 
sure engines  really  advantageous  results,  it  is 
necessary  that  the  fall  of  caloric  should  be  most 
profitably  utilized.  It  is  not  enough  that  the  steam 
be  produced  at  a  high  temperature  :  it  is  also 
necessary  that  by  the  expansion  of  its  volume 
its  temperature  should  become  sufficiently  low.  A 
good  steam-engine,  therefore,  should  not  only  em- 
ploy steam  under  heavy  pressure,  but  under  succes- 
sive and  very  variable  pressures,  differ- 
ing greatly  from  one  another,  and  pro- 
gressively decreasing.* 

In  order  to  understand  in  some  sort 
a  posteriori  the  advantages  of  high- 
pressure  engines,  let  us  suppose  steam 
to  be  formed  under  atmospheric 
pressure  and  introduced  into  the  cylin- 
drical vessel  abed  (Fig.  5),  under  the 
piston  cd,  which  at  first  touches  the 
bottom  ab.  The  steam,  after  having  FIG.  5. 
moved  the  piston  from  ab  to  cd,  will  continue 

*Note  G,  Appendix  B. 


116  MOTIVE  POWER  OF  HEAT. 

finally  to  produce  its  results  in  a  manner  with 
which  we  will  not  concern  ourselves. 

Let  us  suppose  that  the  piston  having  moved  to  cd 
is  forced  downward  to  ef,  without  the  steam  being 
allowed  to  escape,  or  any  portion  of  its  caloric  to  be 
lost.  It  will  be  driven  back  into  the  space  abef,  and 
will  increase  at  the  same  time  in  density,  elastic 
force,  and  temperature.  If  the  steam,  instead  of 
being  produced  under  atmospheric  pressure,  hud 
been  produced  just  when  it  was  being  forced  back 
into  cibef,  and  so  that  after  its  introduction  into  the 
cylinder  it  had  made  the  piston  move  from  ab  to 
ef,  and  had  moved  it  simply  by  its  extension  of 
volume,  from  ef  to  cd,  the  motive  power  produced 
would  have  been  more  considerable  than  in  the  first 
case.  In  fact,  the  movement  of  the  piston,  while 
equal  in  extent,  would  have  taken  place  under  the 
action  of  a  greater  pressure,  though  variable, 
and  though  progressively  decreasing. 

The  steam,  however,  would  have  required  for  its 
formation  exactly  the  same  quantity  of  caloric,  only 
the  caloric  would  have  been  employed  at  a  higher 
temperature. 

It  is  considerations  of  this  nature  which  have  led 
to  the  making  of  double-cylinder  engines — engines 
invented  by  Mr.  Hornblower,  improved  by  Mr. 
Woolf,  and  which,  as  regards  economy  of  the  com- 


MOTIVE  POWER  OF  HEAT.  117 

bustible,  are  considered  the  best.  They  consist  of 
a  small  cylinder,  which  at  each  pulsation  is  filled 
more  or  less  .(often  entirely)  with  steam,  and  of  a 
second  cylinder  having  usually  a  capacity  quadruple 
that  of  the  first,  and  which  receives  no  steam  ex- 
cept that  which  has  already  operated  in  the  first 
cylinder.  Thus  the  steam  when  it  ceases  to  act 
has  at  least  quadrupled  in  volume.  From  the 
second  cylinder  it  is  carried  directly  into  the  con- 
denser, but  it  is  conceivable  that  it  might  be  carried 
into  a  third  cylinder  quadruple  the  second,  and  in 
which  its  volume  would  have  become  sixteen  times 
the  original  volume.  The  principal  obstacle  to  the 
use  of  a  third  cylinder  of  this  sort  is  the  capacity 
which  it  would  be  necessary  to  give  it,  and  the  large 
dimensions  which  the  openings  for  the  passage  of 
the  steam  must  have.  We  will  say  no  more  on  this 
subject,  as  we  do  not  propose  here  to  enter  into  the 
details  of  construction  of  steam-engines.  These 
details  call  for  a  work  devoted  specially  to  them, 
and  which  does  not  yet  exist,  at  least  in  France.* 

*  We  find  in  the  work  called  De  la  Eichesse  Minerals,  by 
M.  Heron  de  Villefosse,  vol.  iii.  p.  50  and  following,  a 
good  description  of  the  steam-engines  actually  in  use  in 
mining.  In  England  the  steam-engine  has  been  very  fully 
discussed  in  the  Encyclopedia  Britannica.  Some  of  the 
data  here  employed  are  drawn  from  the  latter  work. 


118  MOTIVE  POWER  OF  HEAT. 

If  the  expansion  of  the  steam  is  mainly  limited 
by  the  dimensions  of  the  vessels  in  which  the  dila- 
tation must  take  place,  the  degree  of  condensation 
at  which  it  is  possible  to  use  it  at  first  is  limited 
only  by  the  resistance  of  the  vessels  in  which  it  is 
produced,  that  is,  of  the  boilers. 

In  this  respect  we  have  by  no  means  attained 
the  best  possible  results.  The  arrangement  of  the 
boilers  generally  in  use  is  entirely  faulty,  although 
the  tension  of  the  steam  rarely  exceeds  from  four 
to  six  atmospheres.  They  often  burst  and  cause 
severe  accidents.  It  will  undoubtedly  be  possible 
to  avoid  such  accidents,  and  meantime  to  raise  the 
steam  to  much  greater  pressures  than  is  usually 
done. 

Besides  the  high-pressure  double-cylinder  en- 
gines of  which  we  have  spoken,  there  are  also  high- 
pressure  engines  of  one  cylinder.  The  greater  part 
of  these  latter  have  been  constructed  by  two  in- 
genious English  engineers,  Messrs.  Trevithick  and 
Vivian.  They  employ  the  steam  under  a  very  high 
pressure,  sometimes  eight  to  ten  atmospheres,  but 
they  have  no  condenser.  The  steam,  after  it  has 
been  introduced  into  the  cylinder,  undergoes 
therein  a  certain  increase  of  volume,  but  preserves 
always  a  pressure  higher  than  atmospheric.  When 
it  has  fulfilled  its  office  it  is  thrown  out  into  the 


MOTIVE  POWER  OF  HEAT.  119 

atmosphere.  It  is  evident  that  this  mode  of  work- 
ing is  fully  equivalent,  in  respect  to  the  motive 
power  produced,  to  condensing  the  steam  at  100°, 
and  that  a  portion  of  the  useful  effect  is  lost.  But 
the  engines  working  thus  dispense  with  condenser 
and  air-pump.  They  are  less  costly  than  the 
others,  less  complicated,  occupy  less  space,  and  can 
be  used  in  places  where  there  is  not  sufficient  water 
for  condensation.  In  such  places  they  are  of  in- 
estimable advantage,  since  no  others  could  take 
their  place.  These  engines  are  principally  em- 
ployed in  England  to  move  coal-wagons  on  rail- 
roads laid  either  in  the  interior  of  mines  or  outside 
of  them. 

We  have,  further,  only  a  few  remarks  to  make 
upon  the  use  of  permanent  gases  and  other  vapors 
than  that  of  water  in  the  development  of  the  mo- 
tive power  of  heat. 

Various  attempts  have  been  made  to  produce 
motive  power  by  the  action  of  heat  on  atmospheric 
air.  This  gas  presents,  as  compared  with  vapor  of 
water,  both  advantages  and  disadvantages,  which 
we  will  proceed  to  examine. 

(1)  It  presents,  as  compared  with  vapor  of  water, 
a  notable  advantage  in  that,  having  for  equal  vol- 
ume a  much  less  capacity  for  heat,  it  would  cool 
more  rapidly  by  an  equal  increase  of  vohime. 


120  MOTIVE  POWER  OF  HEAT. 

(This  fact  is  proved  by  what  has  already  been 
stated.)  Now  we  have  seen  how  important  it  is  to 
produce  by  change  of  volume  the  greatest  possible 
changes  of  temperature. 

(2)  Vapors  of  water  can  be  formed  only  through 
the  intervention  of  a  boiler,  while  atmospheric  air 
could  be  heated  directly  by  combustion  carried  on 
within  its  own  mass.     Considerable  loss  could  thus 
be  prevented,  not  only  in  the  quantity  of  heat,  but 
also  in  its  temperature.     This  advantage  belongs 
exclusively  to   atmospheric  air.     Other  gases  do 
not  possess  it.     They  would  be  even  more  difficult 
to  heat  than  vapor  of  water. 

(3)  In  order  to   give  to  air  great  increase   of 
volume,  and  by  that  expansion  to  produce  a  great 
change  of  temperature,  it  must  first  be  taken  under 
a  sufficiently  high  pressure;  then  it  must  be  com- 
pressed with  a  pump  or  by  some  other  means  be- 
fore heating  it.     This  operation  would  require  a 
special  apparatus,  an  apparatus  not  found  in  steam- 
engines.     In  the  latter,  water  is  in  a  liquid  state 
when  injected  into  the  boiler,  and  to  introduce  it 
requires  but  a  small  pump. 

(4)  The  condensing  of  the  vapor  by  contact  with 
the  refrigerant  body  is   much  more  prompt  and 
much  easier  than  is  the  cooling   of   air.     There 
might,  of  course,  be  the  expedient  of  throwing  the 


MOTIVE  POWER  OF  HEAT,  121 

latter  out  into  the  atmosphere,  and  there  would  be 
also  the  advantage  of  avoiding  the  use  of  a  refrig- 
erant, which  is  not  always  available,  but  it  would  be 
requisite  that  the  increase  of  the  volume  of  the  air 
should  not  reduce  its  pressure  below  that_  of  the 
atmosphere. 

(5)  One  of  the  gravest  inconveniences  of  steam 
is  that  it  cannot  be  used  at  high  temperatures  with- 
out necessitating  the  use  of  vessels  of  extraordinary 
strength.  It  is  not  so  with  air  for  which  there  ex- 
ists no  necessary  relation  between  the  elastic  force 
and  the  temperature.  Air,  then,  would  seem  more 
suitable  than  steam  to  realize  the  motive  power  of 
falls  of  caloric  from  high  temperatures.  Perhaps 
in  low  temperatures  steam  may  be  more  conven- 
ient. "We  might  conceive  even  the  possibility  of 
making  the  same  heat  act  successively  upon  air  and 
vapor  of  water.  It  would  be  only  necessary  that 
the  air  should  have,  after  its  use,  an  elevated  tem- 
perature, and  instead  of  throwing  it  out  immedi- 
ately into  the  atmosphere,  to  make  it  envelop  a 
steam-boiler,  as  if  it  issued  directly  from  a 
furnace. 

The  use  of  atmospheric  air  for  the  development 
of  the  motive  power  of  heat  presents  in  practice 
very  great,  but  perhaps  not  insurmountable,  diffi- 
culties. If  we  should  succeed  in  overcoming  them, 


122  MOTIVE  POWER  OF  HEAT. 

it  would  doubtless  offer  a  notable  advantage  over 
vapor  of  water.* 

As  to  the  other  permanent  gases,  they  should  be 
absolutely  rejected.  They  have  all  the  inconven- 
iences of  atmospheric  air,  with  none  of  its  advan- 
tages. The  same  can  be  said  of  other  vapors  than 
that  of  water,  as  compared  with  the  latter. 

If  we  could  find  an  abundant  liquid  body  which 
would  vaporize  at  a  higher  temperature  than  water, 
of  which  the  vapor  would  have,  for  the  same  vol- 
ume, a  less  specific  heat,  which  would  not  attack 
the  metals  employed  in  the  construction  of  ma- 
chines, it  would  undoubtedly  merit  the  preference. 
But  nature  provides  no  such  body. 

The  use  of  the  vapor  of  alcohol  has  been  pro- 
posed. Machines  have  even  been  constructed  for  the 
purpose  of  using  it,  by  avoiding  the  mixture  of  ita 
vapor  with  the  water  of  condensation,  that  is,  by 
applying  the  cold  body  externally  instead  of  intro- 
ducing it  into  the  machine.  It  has  been  thought 
that  a  remarkable  advantage  might  be  secured  by 
using  the  vapor  of  alcohol  in  that  it  possesses  a 
stronger  tension  than  the  vapor  of  water  at  the 
same  temperature.  We  can  see  in  this  only  a  fresh 
obstacle  to  be  overcome.  The  principal  defect  of 

*  Note  I,  Appendix  B, 


MOTIVE  POWER  OF  HEAT.  123 

the  vapor  of  water  is  its  excessive  tension  at  an 
elevated  temperature ;  now  this  defect  exists  still 
more  strongly  in  the  vapor  of  alcohol.  As  to  the 
relative  advantage  in  a  greater  production  of  mo- 
tive power, — an  advantage  attributed  to  it, — we 
know  by  the  principles  above  demonstrated  that  it 
is  imaginary. 

It  is  thus  upon  the  use  of  atmospheric  air  and 
vapor  of  water  that  subsequent  attempts  to  perfect 
heat-engines  should  be  based.  It  is  to  utilize  by 
means  of  these  agents  the  greatest  possible  falls  of 
caloric  that  all  efforts  should  be  directed. 

Finally,  we  will  show  how  far  we  are  from  having 
realized,  by  any  means  at  present  known,  all  the 
motive  power  of  combustibles. 

One  kilogram  of  carbon  burnt  in  the  calorimeter 
furnishes  a  quantity  of  heat  capable  of  raising  one 
degree  Centigrade  about  7000  kilograms  of  water, 
that  is,  it  furnishes  7000  units  of  heat  according  to 
the  definition  of  these  units  given  on  page  100. 

The  greatest  fall  of  caloric  attainable  is  measured 
by  the  difference  between  the  temperature  pro- 
duced by  combustion  and  that  of  the  refrigerant 
bodies.  It  is  difficult  to  perceive  any  other  limits 
to  the  temperature  of  combustion  than  those  in 
which  the  combination  between  oxygen  and  the 
combustible  may  take  place.  Let  us  assume,  how- 


124  MOTIVE  POWER  OF  HEAT. 

ever,  that  1000°  may  be  this  limit,  and  we  shall 
certainly  be  below  the  truth.  As  to  the  tempera- 
ture of  the  refrigerant,  let  us  suppose  it  0°.  We 
estimated  approximately  (page  104)  the  quantity  of 
motive  power  that  1000  units  of  heat  develop  be- 
tween 100°  and  99°.  We  found  it  to  be  1. 112  units 
of  power,  each  equal  to  1  metre  of  water  raised  to 
a  height  of  1  metre. 

If  the  motive  power  were  proportional  to  the 
fall  of  caloric,  if  it  were  the  same  for  each  ther- 
mometric  degree,  nothing  would  be  easier  than  to 
estimate  it  from  1000°  to  0°.  Its  value  would  be 

1.112  X  1000  =  1112. 

But  as  this  law  is  only  approximate,  and  as  pos- 
sibly it  deviates  much  from  the  truth  at  high  tem- 
peratures, we  can  only  make  a  very  rough  estimate. 
We  will  suppose  the  number  1112  reduced  one-half, 
that  is,  to  560. 

Since  a  kilogram  of  carbon  produces  7000  units 
of  heat,  and  since  the  number  560  is  relatively 
1000  units,  it  must  be  multiplied  by  7,  which  gives 

7  X  560  =  3920. 

This  is  the  motive  power  of  1  kilogram  of  carbon. 
In  order  to  compare  this  theoretical  result  with 


MOTIVE  POWER  OF  HEAT. 


that  of  experiment,  let  us  ascertain  how  much  mo- 
tive power  a  kilogram  of  carbon  actually  develops 
in  the  best-known  steam-engines. 

The  engines  which,  up  to  this  time,  have  shown 
the  best  results  are  the  large  double-cylinder  en- 
gines used  in  the  drainage  of  the  tin  and  copper 
mines  of  Cornwall.  The  best  results  that  have 
been  obtained  with  them  are  as  follows  : 

65  millions  of  Ibs.  of  water  have  been  raised  one 
English  foot  by  the  bushel  of  coal  burned  (the 
bushel  weighing  88  Ibs.).  This  is  equivalent  to 
raising,  by  a  kilogram  of  coal,  195  cubic  metres  of 
water  to  a  height  of  1  metre,  producing  thereby 
195  units  of  motive  power  per  kilogram  of  coal 
burned. 

195  units-  are  only  the  twentieth  of  3920,  the 
theoretical  maximum  ;  consequently  ^  only  of  the 
motive  power  of  the  combustible  has  been  util- 
ized. 

We  have,  nevertheless,  selected  our  example  from 
among  the  best  steam-engines  known. 

Most  engines  are  greatly  inferior  to  these.  The 
old  engine  of  Chaillot,  for  example,  raised  twenty 
cubic  metres  of  water  thirty-three  metres,  for 
thirty  kilograms  of  coal  consumed,  which  amounts 
to  twenty-two  units  of  motive  power  per  kilogram, 
result  nine  times  less  than  that  given  above, 


126  MOTIVE  POWER  OF  HEAT. 

and  one  hundred  and  eighty  times  less  than  the 
theoretical  maximum. 

We  should  not  expect  ever  to  utilize  in  practice 
all  the  motive  power  of  combustibles.  The  at- 
tempts made  to  attain  this  result  would  be  far  more 
hurtful  than  useful  if  they  caused  other  important 
considerations  to  be  neglected.  The  economy  of 
the  combustible  is  only  one  of  the  conditions  to  be 
fulfilled  in  heat-engines.  In  many  cases  it  is  only 
secondary.  It  should  often  give  precedence  to 
safety,  to  strength,  to  the  durability  of  the  engine, 
to  the  small  space  which  it  must  occupy,  to  small 
cost  of  installation,  etc.  To  know  how  to  appreciate 
in  each  case,  at  their  true  value,  the  considerations 
of  convenience  and  economy  which  may  present 
themselves  ;  to  know  how  to  discern  the  more  im- 
portant of  those  which  are  only  accessories  ;  to  bal- 
ance them  properly  against  each  other,  in  order  to 
attain  the  best  results  by  the  simplest  means  :  such 
should  be  the  leading  characteristics  of  the  man 
called  to  direct,  to  co-ordinate  among  themselves  the 
labors  of  his  comrades,  to  make  them  co-operate 
towards  one  useful  end,  of  whatsoever  sort  it  may 
be. 


(To  face  p.  127.) 


IV.* 

CARNOT'S    THEORY  OF   THE  MOTIVE    POWER 
OF  HEAT,  f 

WITH  NUMERICAL  RESULTS  DEDUCED  FROM  REGNAULT'S 
EXPERIMENTS  ON  STEAM.  J 

BY  SIR  WILLIAM  THOMSON  [LORD  KELVIN], 

1.  THE  presence  of  heat  may  be  recognized  in 
every  natural  object  ;  and  there  is  scarcely  an 
operation  in  nature  which  is  not  more  or  less 

*  From  Transactions  of  the  Edinburgh  Royal  Society,  xiv. 
1849  ;  Annales  de  Chimie,  xxxv.  1852. 

f  Published  in  1824,  in  a  work  entitled  "Reflexions  BUT 
la  Puissance  Motrice  du  Feu,  et  sur  les  Machines  Propres  d 
Developer  cette  Puissance.  Par  S.  Car  not."  [Note  of  Nov. 
5,  1881.  The  original  work  has  now  been  republished, 
with  a  biographical  notice,  Paris,  1878.] 

\  An  account  of  the  first  part  of  a  series  of  researches 
undertaken  by  Mons.  Regnault,  by  order  of  the  late 
French  Government,  for  ascertaining  the  various  physical 
data  of  importance  in  the  theory  of  the  steam-engine,  has 


128  THOMSON  ON  CARNOT'S 

affected  by  its  all-pervading  influence.  An  evolu- 
tion and  subsequent  absorption  of  heat  generally 
give  rise  to  a  variety  of  effects  ;  among  which  may 
be  enumerated,  chemical  combinations  or  decom- 
positions ;  the  fusion  of  solid  substances ;  the 
vaporization  of  solids  or  liquids  ;  alterations  in  the 
dimensions  of  bodies,  or  in  the  statical  pressure 
by  which  their  dimensions  may  be  modified  ;  me- 
chanical resistance  overcome ;  electrical  currents 
generated.  In  many  of  the  actual  phenomena  of 
nature  several  or  all  of  these  effects  are  produced 
together ;  and  their  complication  will,  if  we 
attempt  to  trace  the  agency  of  heat  in  producing 
any  individual  effect,  give  rise  to  much  perplex- 
ity. It  will,  therefore,  be  desirable,  in  laying  the 
foundation  of  a  physical  theory  of  a-ny  of  the 
effects  of  heat,  to  discover  or  to  imagine  phe- 
nomena free  from  all  such  complication,  and  de- 
pending on  a  definite  thermal  agency  ;  in  which 
the  relation  between  the  cause  and  effect,  traced 

been  recently  published  (under  the  title  "  Relation  des 
Experiences,"  etc.)  in  the  Memoires  de  I'Institut,  of  which 
it  constitutes  the  twenty-first  volume  (1847).  The  second 
part  of  these  researches  has  not  yet  been  published.  [Note 
of  Nov.  5,  1881.  The  continuation  of  these  researches  has 
now  been  published  ;  thus  we  have  for  the  whole  series, 
vol.  i.  in  1847  ;  vol.  ii.  in  1862  ;  and  vol.  iii.  in  1870.] 


MOTIVE  POWER  OF  HEAT.  129 

through  the  medium  of  certain  simple  operations, 
may  be  clearly  appreciated.  Thus  it  is  that 
Carnot,  in  accordance  with  the  strictest  principles 
of  philosophy,  enters  upon  the  investigation  of  the 
theory  of  the  motive  power  of  heat. 

2.  The  sole  effect  to  be  contemplated  in  inves- 
tigating the  motive  power  of    heat  is  resistance 
overcome,  or,  as  it  is  frequently  called,  "  work  per- 
formed" or  "  mechanical  effect"     The  questions  to 
be  resolved  by  a  complete  theory  of  the  subject  are 
the  following: 

(1)  What  is  the  precise  nature  of  the  thermal 
agency  by  means  of  which  mechanical  effect  is  to 
be  produced,  without  effects  of  any  other  kind? 

(2)  How    may    the    amount    of    this    thermal 
agency  necessary  for  performing  a  given  quantity 
of  work  be  estimated? 

3.  In  the  following  paper  I  shall  commence  by 
giving  a  short  abstract  of  the  reasoning  by  which 
Carnot  is  led  to  an  answer  to  the  first  of  these 
questions  ;    I  shall  then  explain  the  investigation 
by  which,  in  accordance  with  his  theory,  the  ex- 
perimental elements  necessary  for  answering  the 
second  question  are  indicated  ;  and,  in  conclusion, 
I  shall  state  the  data  supplied  by  Regnault's  recent 
observations  on  steam,  and  apply  them  to  obtain, 
as  approximately  as  the  present  state  of  experi- 


130  THOMSON  ON  CARNOT'8 

mental  science  enables  us  to  do,  a  complete  solu- 
tion of  the  question. 

I.  On  the  nature  of  Thermal  agency,  considered 
as  a  motive  power. 

4.  There  are  [at  present  known]  two,  and  only 
two,  distinct  ways  in  which  mechanical  effect  can 
be  obtained  from  heat.     One  of  these  is  by  means 
of  the  alterations  of  volume,  which  bodies  may  ex- 
perience through  the  action  of  heat ;  the  other  is 
through  the  medium  of  electric  agency.     Seebeck's 
discovery  of  thermo-electric  currents  enables  us  at 
present  to  conceive  of  an  electro-magnetic  engine 
supplied  from  a  thermal  origin,  being  used  as  a 
motive  power ;  but  this  discovery  was  not  made 
until  1821,  and  the  subject  of  thermo-electricity 
can  only  have  been  generally  known  in  a  few  iso- 
lated facts,  with  reference  to  the  electrical  effects 
of  heat  upon  certain  crystals,  at  the  time  when 
Carnot  wrote.     He   makes  no  allusion  to  it,  but 
confines   himself  to   the    method    for    rendering 
thermal  agency  available  as  a  source  of  mechanical 
effect,  by  means  of  the  expansions  and  contrac- 
tions of  bodies. 

5.  A  body  expanding  or  contracting  under  the 
action  of  force  may,  in  general,  either  produce 
mechanical  effect  by  overcoming  resistance,  or  re- 
ceive mechanical  effect  by  yielding  to  the  action 


MOTIVE  POWER  OF  HEAT.  131 

of  force.  The  amount  of  mechanical  effect  thus 
developed  will  depend  not  only  on  the  calorific 
agency  concerned,  but  also  on  the  alteration  in  the 
physical  condition  of  the  body.  Hence,  after  al- 
lowing the  volume  and  temperature  of  the  body  to 
change,  we  must  restore  it  to  its  original  tempera- 
ture and  volume;  and  then  we  may  estimate  the 
aggregate  amount  of  mechanical  effect  developed 
as  due  solely  to  the  thermal  origin. 

6.  Now  the  ordinarily-received,  and  almost  uni- 
versally-acknowledged, principles  with  reference 
to  "quantities  of  caloric"  and  "latent  heat"  lead 
us  to  conceive  that,  at  the  end  of  a  cycle  of  opera- 
tions, when  a  body  is  left  in  precisely  its  primitive 
physical  condition,  if  it  has  absorbed  any  heat  dur- 
ing one  part  of  the  operations,  it  must  have  given 
out  again  exactly  the  same  amount  during  the  re- 
mainder of  the  cycle.  The  truth  of  this  principle 
is  considered  as  axiomatic  by  Carnot,  who  admits 
it  as  the  foundation  of  his  theory  ;  and  expresses 
himself  in  the  following  terms  regarding  it,  in  a 
note  on  one  of  the  passages  of  his  treatise  :* 

"  In  our  demonstrations  we  tacitly  assume  that 
after  a  body  has  experienced  a  certain-  number  of 
transformations,  if  it  be  brought  identically  to  its 

*  Carnot,  p.  67. 


THOMSON  ON  CARNOT'8 


primitive  physical  state  as  to  density,  temperature, 
and  molecular  constitution,  it  must  contain  the 
same  quantity  of  heat  as  that  which  it  initially  pos- 
sessed; or,  in  other  words,  we  suppose  that  the 
quantities  of  heat  lost  by  the  body  under  one  set 
of  operations  are  precisely  compensated  by  those 
which  are  absorbed  in  the  others.  This  fact  has 
never  been  doubted  ;  it  has  at  first  been  admitted 
without  reflection,  and  afterwards  verified,  in  many 
cases,  by  calorimetrical  experiments.  To  deny  it 
would  be  to  overturn  the  whole  theory  of  heat,  in 
which  it  is  the  fundamental  principle.  It  must  be 
admitted,  however,  that  the  chief  foundations  on 
which  the  theory  of  heat  rests,  would  require  a 
most  attentive  examination.  Several  experimental 
facts  appear  nearly  inexplicable  in  the  actual  state 
of  this  theory." 

7.  Since  the  time  when  Carnot  thus  expressed 
himself,  the  necessity  of  a  most  careful  examina- 
tion of  the  entire  experimental  basis  of  the  theory 
of  heat  has  become  more  and  more  urgent.  Es- 
pecially all  those  assumptions  depending  on  the 
idea  that  heat  is  a  substance,  invariable  in  quan- 
tity; not  convertible  into  any  other  element,  and 
incapable  of  being  generated  by  any  physical 
agency;  in  fact  the  acknowledged  principles  of 
latent  heat,—  would  require  to  be  tested  by  a  most 


MOTIVE  POWER  OF  HEAT.  133 

searching  investigation  before  they  ought  to  be 
admitted,  as  they  usually  have  been,  by  almost 
every  one  who  has  been  engaged  on  the  subject, 
whether  in  combining  the  results  of  experimental 
research,  or  in  general  theoretical  investigations. 

8.  The  extremely  important  discoveries  recently 
made  by  Mr.  Joule  of  Manchester,  that  heat  is 
evolved  in  every  part  of  a  closed  electric  conductor, 
moving  in  the  neighborhood  of  a  magnet,*  and 

*  The  evolution  of  heat  in  a  fixed  conductor,  through 
which  a  galvanic  current  is  sent  from  any  source  whatever, 
has  long  been  known  to  the  scientific  world ;  but  it  was 
pointed  out  by  Mr.  Joule  that  we  cannot  infer  from  any 
previously-published  experimental  researches,  the  actual 
generation  of  heat  when  the  current  originates  in  electro- 
magnetic induction;  since  the  question  occurs,  is  the  lieat 
which  is  evolved  in  one  part  of  the  closed  conductor  merely 
transferred  from  tJiose  parts  which  are  subject  to  the  inducing 
influence  ?  Mr.  Joule,  after  a  most  careful  experimental 
investigation  with  reference  to  this  question,  finds  that  it 
must  be  answered  in  the  negative.  (See  a  paper  "On  the 
Calorific  Effects  of  Magneto-Electricity,  and  on  the  Me- 
chanical Value  of  Heat;  by  J.  P.  Joule,  Esq."  Read  be- 
fore the  British  Association  at  Cork  in  1843,  and  subse- 
quently communicated  by  the  Author  to  the  Philosophical 
Magazine,  vol.  xxiii.,  pp.  263,  347,  435.) 

Before  we  can  finally  conclude  that  heat  is  absolutely 
generated  in  such  operations,  it  would  be  necessary  to 
prove  that  the  inducing  magnet  does  not  become  lower  in 


134  THOMSON  ON  CARNOT'S 

that  heat  is  generated  by  the  friction  of  fluids  in 
motion,  seem  to  overturn  the  opinion  commonly 
held  that  heat  cannot  be  generated,  but  only  pro- 
duced from  a  source,  where  it  has  previously  ex- 
isted either  in  a  sensible  or  in  a  latent  condition. 

In  the  present  state  of  science,  however,  no  opera- 
tion is  known  by  which  heat  can  be  absorbed  into 
a  body  without  either  elevating  its  temperature  or 
becoming  latent,  and  producing  some  alteration  in 
its  physical  condition;  and  the  fundamental  axiom 
adopted  by  Carnot  may  be  considered  as  still  the 
most  probable  basis  for  an  investigation  of  the  mo- 
tive power  of  heat;  although  this,  and  with  it 
every  other  branch  of  the  theory  of  heat,  may 
ultimately  require  to  be  reconstructed  upon  another 
foundation,  when  our  experimental  data  are  more 
complete.  On  this  understanding,  and  to  avoid  a 


temperature,  and  thus  compensate  for  the  heat  evolved  in 
the  conductor.  I  am  not  aware  that  any  examination  with 
reference  to  the  truth  of  this  conjecture  has  been  instituted ; 
but,  in  the  case  where  the  inducing  body  is  a  pure  electro- 
magnet (without  any  iron),  the  experiments  actually  per- 
formed by  Mr.  Joule  render  the  conclusion  probable  that 
the  heat  evolved  in  the  wire  of  the  electro-magnet  is  not 
affected  by  the  inductive  action,  otherwise  than  through 
the  reflected  influence  which  increases  the  strength  of  its 
own  current. 


MOTIVE  POWER  OF  HEAT.  135 

repetition  of  doubts,  I  shall  refer  to  Carnot's  funda- 
mental principle,  in  all  that  follows,  as  if  its  truth 
were  thoroughly  established. 

9.  We  are  now  led  to  the  conclusion  that  the 
origin  of  motive  power,  developed  by  the  alternate 
expansions  and  contractions  of  a  body,  must  be 
found  in  the  agency  of  heat  entering  the  body  and 
leaving  it ;  since  there  cannot,  at  the  end  of  a  com- 
plete cycle,  when  the  body  is  restored  to  its  primi- 
tive physical  condition,  have  been  any  absolute  ab- 
sorption of  heat,  and  consequently  no  conversion 
of  heat,  or  caloric,  into  mechanical  effect;  and  it 
remains  for  us  to  trace  the  precise  nature  of  the 
circumstances  under  which  heat  must  enter  the 
body,  and  afterwards  leave  it,  so  that  mechanical 
effect  may  be  produced.     As  an  example,  we  may 
consider  that  machine  for  obtaining  motive  power 
from  heat   with  which  we  are  most  familiar— the 
steam-engine. 

10.  Here,  we  observe,  that  heat  enters  the  ma- 
chine from  the  furnace,  through  the  sides  of  the 
boiler,  and  that  heat  is  continually  abstracted  by 
the  water  employed  for  keeping  the  condenser  cool. 
According  to  Carnot's  fundamental  principle,  the 
quantity  of  heat  thus  discharged,  during  a  complete 
revolution  (or  double  stroke)  of  the  engine,  must  be 
precisely  equal  to  that  which  enters  the  water  of 


136  THOMSON  ON  CARNOT'S 

the  boiler;*  provided  the  total  mass  of  water  and 
steam  be  invariable,  and  be  restored  to  its  primitive 
physical  condition  (which  will  be  the  case  rigorously, 
if  the  condenser  be  kept  cool  by  the  external  appli- 
cation of  cold  water  instead  of  by  injection,  as  is 
more  usual  in  practice),  and  if  the  condensed 
water  be  restored  to  the  boiler  at  the  end  of  each 
complete  revolution.  Thus  we  perceive  that  a  cer- 
tain quantity  of  heat  is  let  down  from  a  hot  body, 
the  metal  of  the  boiler,  to  another  body  at  a  lower 
temperature,  the  metal  of  the  condenser;  and  that 
there  results  from  this  transference  of  heat  a  certain 
development  of  mechanical  effect. 

11.  If  we  examine  any  other  case  in  which 
mechanical  effect  is  obtained  from  a  thermal  origin, 
by  means  of  the  alternate  expansions  and  contrac- 
tions of  any  substance  whatever,  instead  of  the 
water  of  a  steam-engine,  we  find  that  a  similar 
transference  of  heat  is  effected,  and  we  may  there- 
fore answer  the  first  question  proposed,  in  the  fol- 
lowing manner : 

The  thermal  agency  ~by  which  mechanical  effect 
may  be  obtained  is  the  transference  of  heat  from 
one  body  to  another  at  a  lower  temperature. 

*  So  generally  is  Carnot's  principle  tacitly  admitted  as  an 
axiom,  that  its  application  in  this  case  has  never,  so  far  as 
I  am  aware,  been  questioned  by  practical  engineers.  (1849). 


MOTIVE  POWER  OF  HEAT.  137 

11.  On   the   measurement  of  Thermal  Agency, 
considered   with    reference   to    its    equivalent    of 
mechanical  effect. 

12.  A  perfect    thermodynamic    engine    of    any 
kind  is  a  machine  by  means  of  which  the  greatest 
possible  amount  of  mechanical  effect  can  be  obtained 
from  a  given  thermal  agency;  and,  therefore,  if  in 
any  manner  we  can  construct  or  imagine  a  perfect 
engine  which  may  be  applied  for  the  transference 
of  a  given  quantity  of  heat  from  a  body  at  any 
given  temperature  to  another  body  at  a  lower  given 
temperature,  and  if  we  can  evaluate  the  mechanical 
effect  thus  obtained,  we  shall  be  able  to  answer 
the  question  at  present  under  consideration,  and 
so  to  complete  the  theory  of  the   motive  power 
of  heat.     But  whatever  kind  of  engine  we  may 
consider  with  this  view,  it  will  be  necessary  for  us 
to   prove   that  it  is   a  perfect   engine;   since  the 
transference  of  the  heat  from  one  body  to  the  other 
may  be  wholly,  or  partially,  effected  by  conduction 
through   a   solid,*   without    the    development    of 

*When  "  thermal  agency"  is  thus  spent  in  conducting 
heat  through  a  solid,  what  becomes  of  the  mechanical 
effect  which  it  might  produce?  Nothing  can  be  lost  in 
the  operations  of  nature— no  energy  can  be  destroyed. 
What  effect,  then,  is  produced  in  place  of  the  mechanical 
effect  which  is  lost  ?  A  perfect  theory  of  heat  impera- 


138  THOMSON  ON  CARNOT'S 

mechanical  effect;  and,  consequently,  engines  may 
be  constructed  in  which  the  whole  or  any  portion 

tively[demands  an  answer  to  this  question ;  yet  no  answer 
can  be  given  in  the  present  state  of  science.  A  few  years 
ago,  a  similar  confession  must  have  been  made  with  refer- 
ence to  the  mechanical  effect  lost  in  a  fluid  set  in  motion  in 
the  interior  of  a  rigid  closed  vessel,  and  allowed  to  come  to 
rest  by  its  own  internal  friction;  but  in  this  case  the 
foundation  of  a  solution  of  the  difficulty  has  been  ac- 
tually found  in  Mr.  Joule's  discovery  of  the  generation 
of  heat,  by  the  internal  friction  of  a  fluid  in  motion.  En- 
couraged by  this  example,  we  may  hope  that  the  very  per- 
plexing question  in  the  theory  of  heat,  by  which  we  are 
at  present  arrested,  will  before  long  be  cleared  up. 
[Note  of  Sept.,  1881.  The  Theory  of  the  Dissipation  of 
Energy  completely  answers  this  question  and  removes  the 
difficulty.] 

It  might  appear  that  the  difficulty  would  be  entirely 
avoided  by  abandoning  Carnot's  fundamental  axiom  ;  a 
view  which  is  strongly  urged  by  Mr.  Joule  (at  the  conclu- 
sion of  his  paper  "  On  the  Changes  of  Temperature  pro- 
duced by  the  Rarefaction  and  Condensation  of  Air."  Phil. 
Mag.,  May  1845,  vol.  xxvi.)  If  we  do  so,  however,  we 
meet  with  innumerable  other  difficulties— insuperable 
without  farther  experimental  investigation,  and  an  entire 
reconstruction  of  the  theory  of  heat  from  its  foundation. 
It  is  in  reality  to  experiment  that  we  must  look— either 
for  a  verification  of  Carnot's  axiom,  and  an  explanation  of 
the  difficulty  we  have  been  considering;  or  for  an  entirely 
new  basis  of  the  Theory  of  Heat. 


MOTIVE  POWER  OF  HEAT.  139 

of  the  thermal  agency  is  wasted.  Hence  it  is  of 
primary  importance  to  discover  the  criterion  of  a 
perfect  engine.  This  has  been  done  by  Carnot,  who 
proves  the  following  proposition  : 

13.  A  perfect    thermodynamic  engine  is  such 
that,  whatever  amount  of  mechanical  effect  it  can 
derive  from  a  certain  thermal  agency,  if  an  equal 
amount  be  spent  in  working  it  baclcivards,  an  equal 
reverse  thermal  effect  will  be  produced* 

14.  This  proposition  will  be  made  clearer  by  the 
applications  of  it  which  are  given  later  (§  29),  in 
the  cases  of  the  air-engine  and  the  steam-engine, 
than  it  could  be  by  any  general  explanation ;  and  it 
will  also  appear,  from  the   nature  of  the   opera- 
tions described  in  those  cases,  and  the  principles  of 
Carnot's  reasoning,  that  a  perfect  engine  may  be 
constructed  with  any  substance  of  an  indestructible 
texture  as  the  alternately  expanding  and  contract- 
ing medium.      Thus  we  might  conceive   thermo- 
dynamic engines  founded  upon  the  expansions  and 
contractions  of  a  perfectly  elastic  solid,  or  of  a 
liquid;  or  upon  the  alterations  of  volume  experi 
enced  by  substances  in  passing  from  the  liquid  to 
the  solid  state, f  each  of  which  being  perfect,  would 

*  For  a  demonstration,  see  §  29. 

f  A  case  minutely  examined  in  another  paper,  to  be  laid 
before  the  Society  at  the  present  meeting.     ' '  Theoretical 


140  THOMSON  ON  CARNOT'S 

produce  the  same  amount  of  mechanical  effect  from 
a  given  thermal  agency ;  but  there  are  two  cases 
which  Carnot  has  selected  as  most  worthy  of  minute 
attention,  because  of  their  peculiar  appropriateness 
for  illustrating  the  general  principles  of  his  theory, 
no  less  than  on  account  of  their  very  great  practi- 
cal importance:  the  steam-engine,  in  which  the 
substance  employed  as  the  transferring  medium  is 
water,  alternately  in  the  liquid  state  and  in  the 
state  of  vapor  ;  and  the  air-engine,  in  which  the 
transference  is  effected  by  means  of  the  alternate 
expansions  and  contractions  of  a  medium  always 
in  the  gaseous  state.  The  details  of  an  actually 
practicable  engine  of  either  kind  are  not  con- 
templated by  Carnot  in  his  general  theoretical  rea- 
sonings, but  he  confines  himself  to  the  ideal  con- 
struction, in  the  simplest  possible  way  in  each  case, 
of  an  engine  in  which  the  economy  is  perfect.  He 
thus  determines  the  degree  of  perfectibility  which 
cannot  be  surpassed ;  and  by  describing  a  conceiv- 
able method  of  attaining  to  this  perfection  by  an 
air-engine  or  a  steam-engine,  he  points  out  the 
proper  objects  to  be  kept  in  view  in  the  practical 
construction  and  working  of  such  machines.  I  now 
proceed  to  give  an  outline  of  these  investigations. 

Considerations  on  the  Effect  of  Pressure  in  Lowering  the 
Freezing-point  of  Water,"  by  Prof.  James  Thomson. 


MOTIVE  POWER  OF  HEAT.  141 

CARROT'S  THEORY  OF  THE  STEAM-  ENGIKE. 

15.  Let  CDF^E^  be  a  cylinder,  of  which  the 
curved  surface  is  perfectly  impermeable  to  heat, 
with  a  piston  also  impermeable  to  heat,  fitted  in  it ; 
while  the  fixed  bottom  CD,  itself  with  no  capacity 
for  heat,  is  possessed  of  perfect  conducting  power. 
Let  K  be  an  impermeable  stand,  such  that  when 
the  cylinder  is  placed  upon  it  the  contents  below 
the  piston  can  neither  gain  nor  lose  heat.  Let  A 
and  B  be  two  bodies  permanently  retained  at  con- 
stant temperatures,  S°  and  T°,  respectively,  of  which 
the  former  is  higher  than  the  latter.  Let  the  cyl- 
inder, placed  on  the  impermeable  stand,  K,  be  par- 
tially filled  with  water,  at  the  temperature  S,  of  the 
body  Ay  and  (there  being  no  air  below  it)  let  the 
piston  be  placed  in  a  position  EF,  near  the  surface  of 
the  water.  The  pressure  of  the  vapor  above  the 
water  will  tend  to  push  up  the  piston,  and  must 
be  resisted  by  a  force  applied  to  the  piston,*  till 

*  In  all  that  follows,  the  pressure  of  the  atmosphere  on 
the  upper  side  of  the  piston  will  be  included  in  the  applied 
forces,  which,  in  the  successive  operations  described,  are 
sometimes  overcome  by  the  upward  motion,  and  some- 
times yielded  to  in  the  motion  downwards.  It  will  be  un- 
necessary, in  reckoning  at  the  end  of  a  cycle  of  operations, 
to  take  into  account  the  work  thus  spent  upon  the  atmos- 
phere, and  the  restitution  which  has  been  made,  since 
these  precisely  compensate  for  one  another. 


142 


THOMSON  ON  CARNOT'S 


the  commencement  of  the  operations,  which  are 
conducted  in  the  following  manner: 

(1)  The  cylinder  being  placed  on  the  body  A, 


E 


F- 


so  that  the  water  and  vapor  may  be  retained  at  the 
temperature  S9  let  the  piston  rise  any  convenient 


MOTIVE  POWER  OF  HEAT.  143 

height  EE^ ,  to  a  position  E^Fl ,  performing  work 
by  the  pressure  of  the  vapor  below  it  during  its 
ascent. 

[During  this  operation  a  certain  quantity,  H,  of  heat, 
the  amount  of  latent  heat  in  the  fresh  vapor  which  is 
formed,  is  abstracted  from  the  body  A] 

(2)  The  cylinder  being  removed,  and  placed  on 
the  impermeable  stand  K,  let  the  piston  rise  grad- 
ually, till,  when  it  reaches  a  position  E^,  the 
temperature  of  the  water  and  vapor  is  T,  the  same 
as  that  of  the  body  B. 

[During  this  operation  the  fresh  vapor  continually 
formed  requires  heat  to  become  latent  ;  and,  therefore,  as 
the  contents  of  the  cylinder  are  protected  from  any  acces- 
sion of  heat,  their  temperature  sinks.] 

(3)  The  cylinder  being  removed  from  K,  and 
placed  on  B,  let  the  piston  be  pushed  down,  till, 
when  it  reaches  the  position  EZFZ,  the  quantity  of 
heat  evolved  and  abstracted  by  B  amounts  to  that 
which,  during  the  first  operation,  was  taken  from  A. 

[Note  of  Nov.  5, 1881.  The  specification  of  this 
operation,  with  a  view  to  the  return  to  the  primi- 
tive condition,  intended  as  the  conclusion  to  the 
four  operations,  is  the  only  item  in  which  Carnot's 
temporary  and  provisional  assumption  of  the  mate- 
riality of  heat  has  effect.  To  exclude  this  hypothe- 
sis, Prof.  James  Thomson  has  suggested  the  fol* 


144  THOMSON  ON  CARNOT'S 

lowing  corrected  specification  for  the  third  opera- 
tion :  Let  the  piston  be  pushed  down,  till  it  readies 
a  position  E^F^ ,  determined  so  as  to  fulfil  the  con- 
dition,  that  at  the  end  of  the  fourth  operation  the 
primitive  temperature  S  shall  be  reached  :*] 

[During  this  operation  the  temperature  of  the  contents 
of  the  cylinder  is  retained  constantly  at  T°,  and  all  the 
latent  heat  of  the  vapor  which  is  condensed  into  water  at 
the  same  temperature  is  given  out  to  JB.] 

(4)  The  cylinder  being  removed  from  B,  and 
placed  on  the  impermeable  stand,  let  the  piston  be 
pushed  down  from  E^F^  to  its  original  position  EF. 

[During  this  operation,  the  impermeable  stand  prevent- 
ing any  loss  of  heat,  the  temperature  of  the  water  and  air 
must  rise  continually,  till  (since  the  quantity  of  heat 
evolved  during  the  third  operation  was  precisely  equal  to 

*  [Note  of  Nov.  5,  1881.  Maxwell  has  simplified  the 
correction  by  beginning  the  cycle  with  Carnot's  second 
operation,  and  completing  it  through  his  third,  fourth, 
and  first  operations,  with  his  third  operation  nearly  as  fol- 
lows : 

let  the  piston  be  pushed  down  to  any  position  E3F3 ; 
then  Carnot's  fourth  operation  altered  to  the  following': 

let  the  piston  be  pushed  down  from  E3F3  until  the  tem- 
perature reaches  its  primitive  value  8  ; 
and  lastly,  Carnot's  first  operation  altered  to  the  follow- 
ing : 

let  the  piston  rise  to  its  primitive  position.] 


MOTIVE  POWER  OF  HEAT.  145 

that  which  was  previously  absorbed)  at  the  conclusion  it 
reaches  its  primitive  value,  S,  in  virtue  of  Carnot's  funda- 
mental axiom.] 

[Note  of  Nov.  5,  1881.  With  Prof.  James  Thomson's 
correction  of  operation  (3),  the  words  in  virtue  of  "  Car- 
not's Fundamental  Axiom"  must  be  replaced  by  "the 
condition  fulfilled  by  operation  (3),"  in  the  description  of 
the  results  of  operation  (4).] 

16.  At  the  conclusion  of  this  cycle  of  operations  * 
the  total  thermal  agency  has  been  the  letting  down 
of  H  units  of  heat  from  the  body  A,  at  the  tem- 
perature S,  to  B,  at  the  lower  temperature  T\  and 
the  aggregate  of  the  mechanical  effect  has  been  a 
certain  amount  of  work  produced,  since  during  the 
ascent  of  the  piston  in  the  first  and  second  opera- 
tions, the  temperature  of  the  water  and  vapor,  and 
therefore  the  pressure  of  the  vapor  on  the  piston, 
was  on  the  whole  higher  than  during  the  descent, 
in  the  third  and  fourth  operations.  It  remains  for 
us  actually  to  evaluate  this  aggregate  amount  of 
work  performed ;  and  for  this  purpose  the  f ollow- 

*  In  Carnot's  work  some  perplexity  is  introduced  with 
reference  to  the  temperature  of  the  water,  w;hich,  in  the 
operations  he  describes,  is  not  brought  back  exactly  to 
what  it  was  at  the  commencement ;  but  the  difficulty 
which  arises  is  explained  by  the  author.  No  such  difficulty 
occurs  with  reference  to  the  cycle  of  operation  described 
in  the  text,  for  which  I  am  indebted  to  Mons.  Clapeyron. 


146  THOMSON  ON  CARNOT'S 

ing  graphical  method  of  representing  the  mechan- 
ical effect  developed  in  the  several  operations,  taken 
from  Mons.  Clapeyron's  paper,  is  extromely  con- 
venient. 

17.  Let  OX  and  OF  be  two  lines  »,t  right  angles 
to  one  another.  Along  0 X  measure  off  distances 
ON^ ,  -ZVi  JVa ,  N^Ns ,  Na  0,  respectively  proportional 
to  the  spaces  described  by  the  piston  during  the 
four  successive  operations  described  above;  and, 
with  reference  to  these  four  operations  respective- 
ly, let  the  following  constructions  be  made: 

(1)  Along  0  Y  measure  a  length  OA,  to  repre- 
sent the  pressure  of  the  saturated  vapor  at  the 
temperature  Sm,  and  draw  A  A  l  parallel  to  OX,  and 
let  it  meet  an  ordinate  through  N^ ,  in  Al . 

(2)  Draw  a  curve  A^PA  such  that,  if  ON  repre- 
sent, at  any  instant  during  the  second  operation, 
the  distance  of  the  piston  from  its  primitive  posi- 
tion, NP  shall  represent  the  pressure  of  the  vapor 
at  the  same  instant. 

(3)  Through  A^  draw  AZA3  parallel  to  OX,  and 
let  it  meet  an  ordinate  through  Nz  in  A9 . 

(4)  Draw  the  curve  A3A  such  that  the  abscissa 
and  ordinate  of  any  point  in  it  may  represent  re- 
spectively the   distances   of   the   piston   from   its 
primitive  position,  and  the  pressure  of  the  vapor, 
at  each  instant  during  the  fourth  operation.     The 


MOTIVE  POWER  OF  HEAT. 


147 


last  point  of  this  curve  must,  according  to  CarnoVs 
fundamental  principle,  coincide  with  A,  since  the 
piston  is,  at  the  end  of  the  cycle  of  operations, 


again  in  its  primitive  position,  and  the  pressure  of 
the  vapor  is  the  same  as  it  was  at  the  beginning. 

18.  Let  us  now  suppose  that  the  lengths,  ON^ , 
JVjJV,,  N^NI,  and  NtO,  represent  numerically  the 
volumes  of  the  spaces  moved  through  by  the  piston 
during  the  successive  operations.  It  follows  that 
the  mechanical  effect  obtained  during  the  first 
operation  will  be  numerically  represented  by  the 
area  AA^N^O;  that  is,  the  number  of  superficial 
units  in  this  area  will  be  equal  to  the  number  of 
"  foot-pounds"  of  work  performed  by  the  ascend- 
ing piston  during  the  first  operation.  The  work 
performed  by  the  piston  during  the  second  opera- 
tion will  be  similarly  represented  by  the  area 


148  THOMSON  ON  CARNOT'S 

A^A^N^N^.  Again,  during  the  third  operation  a 
certain  amount  of  work  is  spent  on  the  piston^ 
which  will  be  represented  by  the  area  A^A^N^N^  ; 
and  lastly,  during  the  fourth  operation,  work  is 
spent  in  pushing  the  piston  to  an  amount  repre- 
sented by  the  area  A3A  ON3 . 

19.  Hence  the  mechanical  effect  (represented 
by  the  area  OA  A  A^N^)  which  was  obtained  dur- 
ing the  first  and  second  operations,  exceeds  the 
work  (represented  by  N^A^A^AO)  spent  during 
the  third  and  fourth,  by  an  amount  represented 
by  the  area  of  the  quadrilateral  figure  AA1A.2A3  ; 
and,  consequently,  it  only  remains  for  us  to 
evaluate  this  area,  that  we  may  determine  the 
total  mechanical  effect  gained  in  a  complete 
cycle  of  operations.  Now,  from  experimental  data, 
at  present  nearly  complete,  as  will  be  explained 
below,  we  may  determine  the  length  of  the  line 
AAl  for  the  given  temperature  S,  and  a  given  ab- 
sorption H,  of  heat,  during  the  first  operation; 
and  the  length  of  A^AZ  for  the  given  lower  tem- 
perature T,  and  the  evolution  of  the  same  quantity 
of  heat  during  the  fourth  operation:  and  the 
curves  A^PA^,  A3P'A  may  be  drawn  as  graphical 
representations  of  actual  observations.  The  figure 
being  thus  constructed,  its  area  may  be  measured, 
and  we  are,  therefore,  in  possession  of  a  graphical 


MOTIVE  POWER  OF  HEAT.  149 

method  of  determining  the  amount  of  mechanical 
effect  to  be  obtained  from  any  given  thermal 
agency.  As,  however,  it  is  merely  the  area  of  the 
figure  which  it  is  required  to  determine,  it  will  not 
be  necessary  to  be  able  to  describe  each  of  the 
curves  A^PA^,  A^P'A,  but  it  will  be  sufficient  to 
know  the  difference  of  the  abscissas  corresponding 
to  any  equal  ordinates  in  the  two;  and  the  follow- 
ing analytical  method  of  completing  the  problem 
is  the  most  convenient  for  leading  to  the  actual 
numerical  results. 

20.  Draw  any  line  PP'  parallel  to  OX,  meeting 
the  curvilinear  sides  of  the  quadrilateral  in  P  and 
P'.  Let  £  denote  the  length  of  this  line,  and  p 
its  distance  from  OX.  The  area  of  the  figure, 
according  to  the  integral  calculus,  will  be  denoted 
by  the  expression 

/>*, 

e/P3 

where  pl  and  p9  (the  limits  of  integration  indicated 
according  to  Fourier's  notation)  denote  the  lines 
OA  and  NtA3,  which  represent  respectively  the 
pressures  during  the  first  and  third  operations. 
Now,  by  referring  to  the  construction  described 
above,  we  see  that  £  is  the  difference  of  the  volumes 
below  the  piston  at  corresponding  instants  of  the 
second  and  fourth  operations,  or  instants  at  which 


150  THOMSON  ON  CARNOT'S 

the  caturated  steam  and  the  water  in  the  cylinder 
have  the  same  pressure  p,  and  consequently  the 
same  temperature,  which  we  may  denote  by  t. 
Again,  throughout  the  second  operation  the  entire 
contents  of  the  cylinder  possess  a  greater  amount 
of  heat  by  H  units  than  during  the  fourth  ;  and, 
therefore,  at  any  instant  of  the  second  operation 
there  is  as  much  more  steam  as  contains  H  units 
of  latent  heat  than  at  the  corresponding  instant 
of  the  fourth  operation.  Hence  if  k  denote  the 
latent  heat  in  a  unit  of  saturated  steam  at  the 
temperature  t,  the  volume  of  the  steam  at  the  two 

TT 

corresponding  instants  must  differ  by  -T-.  Now,  if 
(T  denote  the  ratio  of  the  density  of  the  steam  to 

TT 

that  of  the  water,  the  volume  -j-  of  steam  will  be 

K 

TT 

formed  from  the  volume  a  -y-  of  water  ;   and  con- 

rC 

sequently  we  have,  for  the  difference  of  volumes  of 
the  entire  contents  at  the  corresponding  instants, 


Hence  the  expression  for  the  area  of  the  quadri- 
lateral figure  becomes 


MOTIVE  POWER  OF  HEAT.          151 

Now,  <r,  k,  and  p,  being  quantities  which  depend 
upon  the  temperature,  may  be  considered  as  func- 
tions of  t;  and  it  will  be  convenient  to  modify  the 
integral  so  as  to  make  t  the  independent  variable. 
The  limits  will  be  from  t  —  T  to  t  —  8,  and,  if  we 
denote  by  M  the  value  of  the  integral,  we  have  the 
expression 

dp 

dt.     .    .     .  (1) 

for  the  total  amount  of  mechanical  effect  gained 
by  the  operations  described  above. 

21.  If  the  interval  of  temperatures  be  extremely 

dp 

small,  —  so  small  that  (1  —  cr)  -r-  will  not  sensibly  vary 

for  values  of  t  between  I7  and  8,  —  the  preceding 
expression  becomes  simply 

dp 

T).    .    .      (2) 


This  might,  of  course,  have  been  obtained  at  once 
by  supposing  the  breadth  of  the  quadrilateral 
figure  AA^A^A  to  be  extremely  small  compared 
with  its  length,  and  then  taking  for  its  area,  as  an 
approximate  value,  the  product  of  the  breadth  into 


152  THOMSON  ON  CARNOT'S 

the  line  A  A 1 ,  or  the  line  A^A^  or  any  line  of  in- 
termediate magnitude. 

The  expression  (2)  is  rigorously  correct  for  any 

dp 

interval  S  —  T,  if  the  mean  value  of  (1  —  <r)-r-  for 

that  interval  be  employed  as  the  coefficient  of 
H(S-T). 

CARNOT'S  THEORY  OF  THE  AIR-ENGINE. 

22.  In  the  ideal  air-engine  imagined  by  Carnot 
four  operations  performed  upon  a  mass  of  air  or 
gas  enclosed  in  a  closed  vessel  of  variable  volume 
constitute  a  complete  cycle,  at  the  end  of  which 
the  medium  is  left  in  its  primitive  physical  condi- 
tion; the  construction  being  the  same  as  that  which 
was  described  above  for  the  steam-engine,  a  body 
A,  permanently  retained  at  the  temperature  8,  and 
B  at  the  temperature  T\  an  impermeable  stand  K\ 
and  a  cylinder  and  piston,  which  in  this  case  con- 
tains a  mass  of  air  at  the  temperature  S,  instead 
of  water  in  the  liquid  state,  at  the  beginning  and 
end  of  a  cycle  of  operations.  The  four  successive 
operations  are  conducted  in  the  following  manner  : 

(1)  The  cylinder  is  laid  on  the  body  A,  so  that 
the  air  in  it  is  kept  at  the  temperature  S;  and  the 
piston  is  allowed  to  rise,  performing  work. 


MOTIVE  POWER  OF  HEAT.  153 

(2)  The  cylinder  is  placed  on  the  impermeable 
stand  K,  so  that  its  contents  can  neither  gain  nor 
lose  heat,  and  the  piston  is  allowed  to  rise  farther, 
still  performing  work,  till  the  temperature  of  the 
air  sinks  to  T. 

(3)  The  cylinder  is  placed  on  B,  so  that  the  air 
is  retained  at  the  temperature  T,  and  the  piston  is 
pushed  down  till  the  air  gives  out  to  the  body  B 
as  much  heat  as  it  had  taken  in  from  A,  during  the 
first  operation. 

[Note  of  Nov.  5,  1881.  To  eliminate  the  assumption  of 
the  materiality  of  heat,  make  Professor  James  Thomson's 
correction  here  also  ;  as  above  in  §  15;  or  take  Maxwell's 
rearrangement  of  the  cycle  described  in  the  foot-note  to 
§  15,  p.  144.] 

(4)  The  cylinder  is  placed  on  K9  so  that  no  more 
heat  can  be  taken  in  or  given  out,  and  the  piston 
is  pushed  down  to  its  primitive  position. 

23.  At  the  end  of  the  fourth  operation  the  tem- 
perature must  have  reached  its  primitive  value  S, 
in  virtue  of  CARNOT'S  axiom. 

24.  Here,  again,  as  in  the  former  case,  we  observe 
that  work  is  performed  by  the  piston  during  the 
first  two  operations ;   and  during  the  third  and 
fourth  work  is  spent  upon  it,  but  to  a  less  amount, 
since  the  pressure  is  on  the  whole  less  during  the 
third  and  fourth  operations  than  during  the  first 


154  THOMSON  ON  CARNOT'8 

and  second,  on  account  of  the  temperature  being 
lower.  Thus,  at  the  end  of  a  complete  cycle  of 
operations,  mechanical  effect  has  been  obtained  ; 
and  the  thermal  agency  from  which  it  is  drawn  is 
the  taking  of  a  certain  quantity  of  heat  from  A, 
and  letting  it  down,  through  the  medium  of  the 
engine,  to  the  body  B  at  a  lower  temperature. 

25.  To  estimate  the  actual  amount  of  effect  thus 
obtained,  it  will  be  convenient  to  consider  the  altera- 
tions of  volume  of  the  mass  of  air  in  the  several 
operations  as  extremely  small.    We  may  afterwards 
pass  by  the   integral  calculus,  or,  practically,  by 
summation   to   determine  the   mechanical    effect 
whatever  be  the  amplitudes  of  the  different  motions 
of  the  piston. 

26.  Let  dq  be  the  quantity  of  heat  absorbed 
during  the  first  operation,  which  is  evolved  again 
during  the  third;  and  let  dv  be  the  corresponding 
augmentation  of  volume  which  takes  place  while 
the  temperature  remains  constant,  as  it  does  during 
the  first  operation.*     The  diminution  of  volume 

*  Thus,  -^  will  be  the  partial  differential  coefficient, 

with  respect  to  ®,  of  that  function  of  wand  t  which  expresses 
the  quantity  of  heat  that  must  be  added  to  a  mass  of  air 
when  in  a  "  standard  "  state  (such  as  at  the  temperature  zero, 
and  under  the  atmospheric  pressure),  to  bring  it  to  the 
temperature  t  and  the  volume  v.  That  there  is  such  a 


MOTIVE  POWER  OF  HEAT.  155 

in  the  third  operation  must  be  also  equal  to  dv,  or 
only  differ  from  it  by  an  infinitely  small  quantity  of 
the  second  order.  During  the  second  operation  we 
may  suppose  the  volume  to  be  increased  by  an  in- 
finitely small  quantity  0;  which  will  occasion  a 
diminution  of  pressure  and  a  diminution  of  tem- 
perature, denoted  respectively  by  GJ  and  r.  During 
the  fourth  operation  there  will  be  a  diminution  of 
volume  and  an  increase  of  pressure  and  temperature, 
which  can  only  differ,  by  infinitely  small  quantities 
of  the  second  order,  from  the  changes  in  the  other 
direction,  which  took  place  in  the  second  operation, 
and  they  also  may,  therefore,  be  denoted  by  0,  GO, 
and  r,  respectively.  The  alteration  of  pressure 

function,  of  two  independent  variables  v  and  t,  is  merely 
an  analytical  expression  of  Carnot's  fundamental  axiom,  as 
applied  to  a  mass  of  air.  The  general  principle  may  be 
analytically  stated  in  the  following  terms  : — If  Mdv  denote 
the  accession  of  heat  received  by  a  mass  of  any  kind,  not 
possessing  a  destructible  texture,  when  the  volume  is  in- 
creased by  dv,  the  temperature  being  kept  constant,  and  if 
Ndt  denote  the  amount  of  heat  which  must  be  supplied  to 
raise  the  temperature  by  dt,  without  any  alteration  of  vol- 
ume ;  then  Mdv -\-Ndt  must  be  the  differential  of  a  func- 
tion of  v  and  t.  [Note  of  Nov.  5, 1881.  In  the  corrected 
theory  it  is  (M  —  Jp)  dv  -\-  Ndttth&t  is  a  complete  differential, 
not  Mdv  +  Ndt.  See  Dynamical  Theory  of  Heat  (Art.  XLVIII.  , 
below),  §  20.  J 


156  THOMSON  ON  CARNOT'S 

during  the  first  and  third  operations  may  at  once 
be  determined  by  means  of  Mariotte's  law,  since 
in  them  the  temperature  remains  constant.  Thus, 
if,  at  the  commencement  of  the  cycle,  the  volume 
and  pressure  be  v  and  p,  they  will  have  become 
v  -f-  dv  and  pv/(v  -f-  civ)  at  the  end  of  the  first 
operation.  Hence  the  diminution  of  pressure 
during  the  first  operation  is  p  —  pv/(v  -f-  dv)  or 
pdv/(v  +  dv)  and  therefore,  if  we  neglect  infinitely 
small  quantities  of  the  second  order,  we  hswepdv/v 
for  the  diminution  of  pressure  during  the  first 
operation  ;  which  to  the  same  degree  of  approxima- 
tion, will  be  equal  to  the  increase  of  pressure  during 
the  third.  If  t  +  T  and  t  be  taken  to  denote  the 
superior  and  inferior  limits  of  temperature,  we 
shall  thus  have  for  the  volume,  the  temperature, 
and  the  pressure  at  the  commencements  of  the 
four  successive  operations,  and  at  the  end  of  the 
cycle,  the  following  values  respectively: 


(1)  v, 

(2)  v  +  dv, 

(3)  v  +  dv+ 

(4)  v  +  0,  t,  p  - 

(5)  v,  t  +  r,    p. 


MOTIVE  POWER  OF  HEAT.  157 

Taking  the  mean  of  the  pressures  at  the  beginning 
and  end  of  each  operation,  we  find 


(4)  j>  -  ^  G*, 

which,  as  we  are  neglecting  infinitely  small  quan- 
tities of  the  second  order,  will  be  the  expressions 
for  the  mean  pressures  during  the  four  successive 
operations.  Now,  the  mechanical  effect  gained  or 
spent,  during  any  of  the  operations,  will  be  found 
by  multiplying  the  mean  pressure  by  the  increase 
or  diminution  of  volume  which  takes  place;  and 
we  thus  find 


(4)  (p  -  \ 


158  THOMSON  ON  CARNOTS 

for  the  amounts  gained  during  the  first  and  second, 
and  spent  during  the  third  and  fourth  operations  ; 
and  hence,  by  addition  and  subtraction,  we  find 

,  ,dv  ,  ,.dv 

codv  —  p4> — ,    or     (VGJ  —  pep) — , 

for  the  aggregate  amount  of  mechanical  effect 
gained  during  the  cycle  of  operations.  It  only  re- 
mains for  us  to  express  this  result  in  terms  of  dq 
and  r,  on  which  the  given  thermal  agency  depends. 
For  this  purpose  we  remark  that  0  and  GO  are  al- 
terations of  volume  and  pressure  which  take  place 
along  with  a  change  of  temperature  r,  and  hence, 
by  the  laws  of  compressibility  and  expansion,  we 
may  establish  a  relation*  between  them  in  the  fol- 
lowing manner : 

Let  p9  be  the  pressure  of  the  mass  of  air  when 
reduced  to  the  temperature  zero,  and  confined 
in  a  volume  v0;  then,  whatever  be  v0 ,  the  product 
pQvQ  will,  by  the  law  of  compressibility,  remain  con- 
stant ;  and,  if  the  temperature  be  elevated  from  0 
to  t  +  *>  and  the  gas  be  allowed  to  expand  freely 
without  any  change  of  pressure,  its  volume  will  be 

*  We  might  also  investigate  another  relation,  to  express 
the  fact  that  there  is  no  accession  or  removal  of  heat  during 
either  the  second  or  the  fourth  operation;  but  it  will  be 
seen  that  this  will  not  affect  the  result  in  the  text,  although 
it  would  enable  us  to  determine  both  0  and  GO  in  terms  of  T. 


OF  THK 

UNIVERSITY 


MOTIVE  POWER  OF  HEAT.  159 

increased  in  the  ratio  of  1  to  1  -|-  E(t  -\-  T),  where 
E  is  very  nearly  equal  to  .00366  (the  Centigrade 
scale  of  the  air-thermometer  being  referred  to), 
whatever  be  the  gas  employed,  according  to  the 
researches  of  Regnault  and  of  Magnus  on  the  ex- 
pansion of  gases  by  heat.  If,  now,  the  volume  be 
altered  arbitrarily  with  the  temperature  continually 
at  t  -f-  7?  the  product  of  the  pressure  and  volume 
will  remain  constant ;  and  therefore  we  have 

pv  =  p0v0{l  +  fi(t  +  r)}. 
Similarly, 

Hence,  by  subtraction,  we  have 
or,  neglecting  the  product  00$, 

Hence  the  preceding  expression  for  mechanical 
effect,  gained  in  the  cycle  of  operations,  becomes 

p0v0 .  Er .  dv/v. 
Or,  as  we  may  otherwise  express  it, 

vdq/dv  '       * 

Hence,  if  we  denote  by  M  the  mechanical  effect  due 
to  H  units  of  heat  descending  through  the  same 
interval  T,  which  might  be  obtained  by  repeating 


160  THOMSON  ON  CARNOT'S 

TT 

the  cycle  of  operations  described  above,  -=-  times, 

we  have  M  =  -^£°-  .  Hr  .....     (3) 

vdq/dv 

27.  If  the  amplitudes  of  the  operations  had  been 
finite,  so  as  to  give  rise  to  an  absorption  of  H  units 
of  heat  during  the  first  operation,  and  a  lowering 
of  temperature  from  8  to  T  during  the  second,  the 
amount  of  work  obtained  would  have  been  found 
to  be  expressed  by  means  of  a  double  definite  in- 
tegral thus  :* 

*=  far  #..&%. 

i/o      ijT          vdq/dv 

or 


this  second  form  being  sometimes  more  convenient. 

*  This  result  might  have  been  obtained  by  applying  the 
usual  ^notation  of  the  integral  calculus  to  express  the 
area  of  the  curvilinear  quadrilateral,  which,  according  to 
Clapeyron's  graphical  construction,  would  be  found  to 
represent  the  entire  mechanical  effect  gained  in  the  cycle 
of  operations  of  the  air-engine.  It  is  not  necessary,  how- 
ever, to  enter  into  the  details  of  this  investigation,  as  the 
formula  (3),  and  the  consequences  derived  from  it,  include 
the  whole  theory  of  the  air-engine,  in  the  best  practical 
form;  and  the  investigation  of  it  which  I  have  given  in  the 
text  will  probably  give  as  clear  a  view  of  the  reasoning  on 
which  it  is  founded  as  could  be  obtained  by  the  graphical 
method,  which  in  this  case  is  not  so  valuable  as  it  is  from 
its  simplicity  in  the  case  of  the  steam-engine. 


MOTIVE  POWER  OF  HEAT.  161 

28.  The  preceding  investigations,  being  founded 
on  the  approximate  laws  of  compressibility  and  ex- 
pansion (known  as  the  law  of  Mariotte  and  Boyle, 
and  the  law  of  Dalton  and  Gay-Lussac),  would  re- 
quire some  slight  modifications  to  adapt  them  to 
cases  in  which  the  gaseous  medium  employed  is  such 
as  to  present  sensible  deviations  from  those  laws. 
Regnault's  very  accurate  experiments  show  that 
the  deviations  are  insensible,  or  very  nearly  so,  for 
the  ordinary  gases  at  ordinary  pressures ;  although 
they  may  be  considerable  for  a  medium,  such  as 
sulphurous  acid,  or  carbonic  acid  under  high  pres- 
sure, which  approaches  the  physical  condition  of  a 
vapor  at  saturation ;  and  therefore,  in  general,  and 
especially  in  practical  applications  to  real  air-engines, 
it  will  be  unnecessary  to  make  any  modification  in 
the  expressions.  In  cases  where  it  may  be  necessary, 
there  is  no  difficulty  in  making  the  modifications, 
when  the  requisite  data  are  supplied  by  experiment. 

29.*  Either  the  steam-engine  or  the  air-engine, 
according  to  the  arrangements  described  above, 
gives  all  the  mechanical  effect  that  can  possibly  be 
obtained  from  the  thermal  agency  employed.  For 

*  This  paragraph  is  the  demonstration,  referred  to  above, 
of  the  proposition  stated  in  §  13,  as  it  is  readily  seen  that 
it  is  applicable  to  any  conceivable  kind  of  therinodynamic 
engine. 


162  THOMSON  ON  CARNOT'S 

it  is  clear  that  in  either  case  the  operations  may 
be  performed  in  the  reverse  order,  with  every 
thermal  and  mechanical  effect  reversed.  Thus,  in 
the  steam-engine,  we  may  commence  by  placing 
the  cylinder  on  the  impermeable  stand,  allow  the 
piston  to  rise,  performing  work,  to  the  position 
E3FS ;  we  may  then  place  it  on  the  body  B,  and 
allow  it  to  rise,  performing  work,  till  it  reaches 
E^F^'y  after  that  the  cylinder  may  be  placed  again 
on  the  impermeable  stand,  and  the  piston  may  be 
pushed  down  to  E^F^ ;  and,  lastly,  the  cylinder 
being  removed  to  the  body  A,  the  piston  may  be 
pushed  down  to  its  primitive  position.  In  this 
inverse  cycle  of  operations  a  certain  amount  of 
work  has  been  spent,  precisely  equal,  as  we  readily 
see,  to  the  amount  of  mechanical  effect  gained  in 
the  direct  cycle  described  above ;  and  heat  has  been 
abstracted  from  B,  and  deposited  in  the  body  A, 
at  a  higher  temperature,  to  an  amount  precisely 
equal  to  that  which  in  the  direct  style  was  let 
down  from  A  to  B.  Hence  it  is  impossible  to 
have  an  engine  which  will  derive  more  mechanical 
effect  from  the  same  thermal  agency  than  is  ob- 
tained by  the  arrangement  described  above;  since, 
if  there  could  be  such  an  engine,  it  might  be  em- 
ployed to  perform,  as  a  part  of  its  whole  work,  the 
inverse  cycle  of  operations,  upon  an  engine  of  the 


MOTIVE  POWER  OF  SEAT  163 

kind  we  have  considered,  and  thus  to  continually 
restore  the  heat  from  B  to  A,  which  has  descended 
from  A  to  B  for  working  itself;  so  that  we  should 
have  a  complex  engine,  giving  a  residual  amount 
of  mechanical  effect  without  any  thermal  agency, 
or  alteration  of  materials,  which  is  an  impossibility 
in  nature.  The  same  reasoning  is  applicable  to 
the  air-engine ;  and  we  conclude,  generally,  that 
any  two  engines,  constructed  on  the  principles  laid 
down  above,  whether  steam-engines  with  different 
liquids,  an  air-engine  and  a  steam-engine,  or  two 
air-engines  with  different  gases,  must  derive  the 
same  amount  of  mechanical  effect  from  the  same 
thermal  agency. 

30.  Hence,  by  comparing  the  amounts  of  me- 
chanical effect  obtained  by  the  steam-engine  and 
the  air-engine  from  the  letting  down  of  the  H 
units  of  heat  from  A  at  the  temperature  (t  -j-  *)  to 
B  at  t,  according  to  the  expressions  (2)  and  (3), 
we  have 

M=(l-a)%L.ffT  =  ^j-.HT..   (5) 

'  kdt  vdq/dv 

If  we  denote  the  coefficient  of  Hr  in  these  equal 
expressions  by  //,  which  maybe  called  '"Carnot's 
coefficient,"  we  have 


164  THOMSON  ON  CARNOT'S 

and  we  deduce  the  following  very  remarkable  con- 
clusions : 

(1)  For  the  saturated  vapors   of   all   different 
liquids,  at  the    same   temperature,  the  value   of 

(1  —  a)  -~-  must  be  the  same. 

(2)  For  any  different  gaseous   masses,  at  the 

same  temperature,  the  value   of      ,  °.  f-  must  be 

vdq/dv 

the  same. 

(3)  The  values  of  these  expressions  for  saturated 
vapors  and  for  gases,  at  the  same  temperature, 
must  be  the  same. 

31.  No  conclusion  can  be  drawn  a  priori  re- 
garding the  values  of  this  coefficient  JJL  for  different 
temperatures,  which  can  only  be  determined,  or 
compared,  by  experiment.  The  results  of  a  great 
variety  of  experiments,  in  different  branches  of 
physical  science  (Pneumatics  and  Acoustics),  cited 
by  Carnot  and  by  Clapeyron,  indicate  that  the 
values  of  JJL  for  low  temperatures  exceed  the  values 
for  higher  temperatures  ;  a  result  amply  verified 
by  the  continuous  series  of  experiments  performed 
by  Regnault  on  the  saturated  vapor  of  water  for  all 
temperatures  from  0°  to  230°,  which,  as  we  shall 
see  later,  give  values  for  /*  gradually  diminishing 
from  the  inferior  limit  to  the  superior  limit  of 


MOTIVE  POWER  OF  HEAT.  165 

temperature.  When,  by  observation,  j*  has  been 
determined  as  a  function  of  the  temperature,  the 
amount  of  mechanical  effect,  M,  deducible  from 
H  units  of  heat  descending  from  a  body  at  the 
temperature  S  to  a  body  at  the  temperature  T, 
may  be  calculated  from  the  expression 

rrt 

M=H  C  pdt,.    .    .     .   (7) 

t/S 

which  is,  in  fact,  what  either  of  the  equations  (1) 
for  the  steam-engine,  or  (4)  for  the  air-engine,  be- 
comes, when  the  notation  //,  for  Carnot's  multi- 
plier, is  introduced. 

The  values  of  this  integral  may  be  practically 
obtained,  in  the  most  convenient  manner,  by  first 
determining,  from  observation,  the  mean  values  of 
/*  for  the  successive  degrees  of  the  thermometric 
scale,  and  then  adding  the  values  for  all  the  de 
grees  within  the  limits  of  the  extreme  temperatures 
tfand  T.* 

32.  The  complete  theoretical  investigation  of 
the  motive  power  of  heat  is  thus  reduced  to  the 
experimental  determination  of  the  coefficient  /t ; 
and  may  be  considered  as  perfect,  when,  by  any 
series  of  experimental  researches  whatever,  we  can 

\ 
*  The  results  of  these  investigations  are  exhibited  in 

Tables  I  and  II. 


166  THOMSON  ON  CARNOT'S 

find  a  value  of  /*  for  every  temperature  within 
practical  limits.  The  special  character  of  the  ex- 
perimental researches,  whether  with  reference  to 
gases  or  with  reference  to  vapors,  necessary  and 
sufficient  for  this  object,  is  defined  and  restricted 
in  the  most  precise  manner,  by  the  expressions  (6) 
for  //,  given  above.* 

33.  The  object  of  Regnault's  great  work,  referred 
to  in  the  title  of  this  paper,  is  the  experimental  de- 
termination of  the  various  physical  elements  of  the 
steam-engine  ;  and  when  it  is  complete,  it  will 
furnish  all  the  data  necessary  for  the  calculation 
of  /*.  The  valuable  researches  already  published 
in  a  first  part  of  that  work  make  known  the 
latent  heat  of  a  given  weight,  and  the  pressure,  of 
saturated  steam  for  all  temperatures  between  0° 
and  230°  Cent,  of  the  air-thermometer.  Besides 
these  data,  however,  the  density  of  saturated  va- 
por must  be  known,  in  order  that  k,  the  latent 
heat  of  a  unit  of  volume,  may  be  calculated  from 
Regnault's  determination  of  the  latent  heat  of  a 
given  weight.  *  Between  the  limits  of  0°  and  100°, 

*  It  is,  comparatively  speaking,  of  little  consequence  to 
know  accurately  the  value  of  or,  for  the  factor  (1  — cr)  of 
the  expression  for  >w,  since  it  is  so  small  (being  less  than 
T^  for  all  temperatures  between  0°  and  100°)  that,  unless 
all  the  data  are  known  with  more  accuracy  than  we  can 


MOTIVE  POWER  OF  HEAT.  167 

it  is  probable,  from  various  experiments  which 
have  been  made,  that  the  density  of  vapor  follows 
very  closely  the  simple  laws  which  are  so  accurately 
verified  by  the  ordinary  gases;*  and  thus  it  may 
be  calculated  from  Regnault's  table  giving  the 
pressure  at  any  temperature  within  those  limits. 
Nothing  as  yet  is  known  with  accuracy  as  to  the 
density  of  saturated  steam  between  100  and  230°, 
and  we  must  be  contented  at  present  to  estimate  it 
by  calculation  from  Regnault's  table  of  pressures; 
although,  when  accurate  experimental  researches 
on  the  subject  shall  have  been  made,  considerable 
deviations  from  the  laws  of  Boyle  and  Dalton,  on 
which  this  calculation  is  founded,  may  be  dis- 
covered. 

34.  Such  are  the  experimental  data  on  which 
the  mean  values  of  //  for  the  successive  degrees  of 
the  air- thermometer,  from  0  to  230°,  at  present 
laid  before  the  Royal  Society,  is  founded.  The 
unit  of  length  adopted  is  the  English  foot;  the 
unit  of  weight,  the  pound  ;  the  unit  of  work,  a 

count  upon  at  present,  we  might  neglect  it  altogether,  and 
take  dp/Mi  simply,  as  the  expression  for  /*,  without  com- 
mitting any  error  of  important  magnitude. 

*  This  is  well  established,  within  the  ordinary  atmos- 
pheric limits,  in  Regnault's  Etudes  Meteorologiques,  in  the 
Annales  de  Chimie,  vol.  xv. ,  1846, 


168  THOMSON  ON  CARNOT'S 

"  foot-pound  ;"and  the  unit  of  heat  that  quantity 
which,  when  added  to  a  pound  of  water  at  0°,  will 
produce  an  elevation  of  1°  in  temperature.  The 
mean  value  of  /*  for  any  degree  is  found  to  a  suffi- 
cient degree  of  approximation  by  taking,  in  place 
of  (r,  dp/dt  and  k  ;  in  the  expression 


the  mean  values  of  those  elements;  or,  what  is 
equivalent  to  the  corresponding  accuracy  of  ap- 
proximation, by  taking,  in  place  of  cr  and  k  respec- 
tively, the  mean  of  the  values  of  those  elements  for 
the  limits  of  temperature,  and  in  place  of  dp/dt, 
the  difference  of  the  values  of  p,  at  the  same  limits. 

35.  In  Regnault's  work  (at  the  end  of  the  eighth 
memoir),  a  table  of  the  pressures  of  saturated  steam 
for  the  successive  temperatures  0°,  1°,  2°,  ...  230°, 
expressed  in  millimetres  of  mercury,  is  given.     On 
account  of  the  units  adopted  in  this  paper,  these 
pressures   must  be   estimated   in   pounds  on  the 
square  foot,  which  we  may  do  by  multiplying  each 
number  of  millirnetre3  by  2.7896,  the  weight  in 
pounds  of  a  sheet  of  mercury,  one  millimetre  thick, 
and  a  square  foot  in  area. 

36.  The  value  of  &,  the  latent  heat  of  a  cubic 
foot,  for  any  temperature  t,  is  found  from  A,  the 


MOTIVE  POWER  OF  HEAT.  169 

latent  heat  of  a  pound  of  saturated  steam,  by  the 
equation 

p        1  +  .00366  X  100  . 
~760      1  +  .00366  X  t 

where  p  denotes  the  pressure  in  millimetres,  and  1 
the  latent  heat  of  a  pound  of  saturated  steam;  the 
values  of  A,  being  calculated  by  the  empirical  for- 
mula f 

A  =  (606.5  -f  0.3050  -(t  +  .00002**  +  0.0000003*'), 
given  by  Regnault  as  representing,  between  the 

*  It  appears  that  the  vol.  of  1  kilog.  must  be  1.69076  ac- 
cording to  the  data  here  assumed. 

The  density  of  saturated  steam  at  100°  is  taken  as  ~^ 
of  that  of  water  at  its  maximum.  Rankine  takes  it  as  T^. 

f  The  part  of  this  expression  in  the  first  vinculum  (see 
Regnault,  end  of  ninth  memoir)  is  what  is  known  as  "  the 
total  heat "  of  a  pound  of  steam,  or  the  amount  of  heat 
necessary  to  convert  a  pound  of  water  at  0°  into  a  pound 
of  saturated  steam  at  t°  ;  which,  according  to  "  Watt's 
law,"  thus  approximately  verified,  would  be  constant. 
The  second  part,  which  would  consist  of  the  single  term 
t,  if  the  specific  heat  of  water  were  constant  for  all  tem- 
peratures, is  the  number  of  thermic  units  necessary  to  raise 
the  temperature  of  a  pound  of  water  from  0°  to  t°,  and 
expresses  empirically  the  results  of  Regnault's  experi- 
ments on  the  specific  heat  of  water  (see  end  of  the  tenth 
memoir),  described  in  the  work  already  referred  to. 


170  THOMSON  ON  CARNOT'S 

extreme  limits  of  his  observations,  the  latent  heat 
of  a  unit  weight  of  saturated  steam. 

EXPLANATION  OF  TABLE  I. 

37.  The  mean  values  of  jn  for  the  first,  for  the 
eleventh,  for  the  twenty-first,  and  so  on,  up  to  the 
231st*  degree  of  the  air-thermometer,  have  been 
calculated  in  the  manner  explained  in  the  preced- 
ing paragraphs.  These,  and  interpolated  results, 
which  must  agree  with  what  would  have  been  ob- 
tained, by  direct  calculation  from  Regnault's  data, 
to  three  significant  places  of  figures  (and  even  for 
the  temperatures  between  0°  and  100°,  the  experi- 
mental data  do  not  justify  us  in  relying  on  any  of 
the  results  to  a  greater  degree  of  accuracy),  are 
exhibited  in  Table  I. 

To  find  the  amount  of  mechanical  effect  due  to  a 
unit  of  heat,  descending  from  a  body  at  a  temper- 
ature 8  to  a  body  at  T,  if  these  numbers  be  in- 
tegers, we  have  merely  to  add  the  values  of  ja  in 
Table  I.  corresponding  to  the  successive  numbers. 

T+\,  T+2,  ....#- 2,  S-l. 

*  In  strictness,  the  230th  is  the  last  degree  for  which  the 
experimental  data  are  complete  ;  but  the  data  for  the  231st 
may  readily  be  assumed  in  a  sufficiently  satisfactory 
manner. 


MOTIVE  POWER  OF  HEAT.  171 


EXPLANATION  OF  TABLE  II. 

38.  The  calculation  of  the  mechanical  effect,  in 
any  case,  which  might  always  be  effected  in  the 
manner  described  in  §  37  (with  the  proper  modifi- 
cation for  fractions  of  degrees,  when  necessary),  is 
much  simplified  by  the  use  of  Table  II.,  where  the 
first  number  of  Table  I.,  the  sum  of  the  first  and 
second,  the  sum  of  the  first  three,  the  sum  of  the 
first  four,  and  so  on,  are  successively  exhibited. 
The  sums  thus  tabulated  are  the  values  of  the  in- 
tegrals 

/I  />2  />3  /*2 

pdt,     I    pdt,     I    »dt,....   I 
e/0  I/O  t/0 

and,  if  we  denote    /  pdt  by  the  letter  M,  Table  II. 

may  be  regarded  as  a  table  of  the  value  of  M. 

To  find  the  amount  of  mechanical  effect  due  to  a 
unit  of  heat  descending  from  a  body  at  a  tempera- 
ture 8  to  a  body  at  T,  if  these  numbers  be  integers, 
we  have  merely  to  subtract  the  value  of  M,  for  the 
number  T,  from  the  value  for  the  number  S,  given 
in  Table  II 


172 


THOMSON  ON  CARNOT'S 


TABLE  I.* 

MEAN  VALUES  OF  n  FOR  THE,  SUCCESSIVE.  DEGREES  OF 
THE  AIR-THERMOMETER  FROM  0°  TO  230°. 


o 

M 

o 

M 

o 

M 

o 

M 

1 

4.960 

32 

4  559 

63 

4.194 

94 

3.889 

2 

4.946 

33 

4.547 

64 

4.183 

95 

3.880 

3 

4.932 

34 

4.535 

65 

4.172 

96 

3.871 

4 

4.918 

35 

4.522 

66 

4.161 

97 

3.863 

5 

4.905 

36 

4.510 

67 

4.150 

98 

3.854 

6 

4.892 

37 

4.498 

68 

4.140 

99 

3.845 

7 

4.878 

38 

4.486 

69 

4.129 

100 

3.837 

8 

4.865 

39 

4.474 

70 

4.119 

101 

3.829 

9 

4.852 

40 

4.462 

71 

4.109 

102 

3.820 

10 

4.839 

41 

4.450 

72 

4.098 

103 

3.812 

11 

4.826 

42 

4.438 

73 

4.088 

104 

3.804 

12 

4.812 

43 

4.426 

74 

4.078 

105 

3.796 

13 

4.799 

44 

4.414 

75 

4.067 

106 

3.788 

14 

4.786 

45 

4.402 

76 

4.057 

107 

3.780 

15 

4.773 

46 

4.390 

77 

4.047 

108 

3.772 

16 

4.760 

47 

4  378 

73 

4.037 

109 

3.764 

17 

4.747 

48 

4.366 

79 

4.028 

110 

3.757 

18 

4  785 

49 

4.355 

80 

4.018 

111 

3.749 

19 

4.722 

50 

4.343 

81 

4.009 

112 

3.741 

20 

4.709 

51 

4.331 

82 

3.999 

113 

3.734 

21 

4.697 

52 

4.319 

83 

3.990 

114 

3.726 

22 

4.684 

53 

4.308 

84 

3.980 

115 

3.719 

23 

4.672 

54 

4.296 

85 

3.971 

116 

3.712 

24 

4.659 

55 

4.285 

86 

3.961 

117 

3.704 

25 

4.646 

56 

4.273 

87 

3.952 

118 

3.697 

26 

4.634 

57 

4.262 

88 

3.943 

119 

3.689 

27 

4.621 

58 

4.250 

89 

3.934 

120 

3.682 

28 

4.609 

59 

4.239 

90  ~ 

3.925 

121 

3.675 

29 

4.596 

60 

4.227 

91 

3.916 

122 

3.668 

30 

4.584 

61 

4.216 

92 

3.907 

123 

3.661 

31 

4.572 

62 

4.205 

93 

3.898 

124 

3.654 

*  The  numbers  here  tabulated  may  also  be  regarded  as 
the  actual  values  ofjufort  =  it  t  =  H,  t  =  2±,  t  =  3|,  etc. 


MOTIVE  POWER  OF  HEAT. 


173 


TABLE  I.— (Continued.} 


0 

p 

0 

» 

0 

M 

0 

P 

125 

3.647 

152 

3.479 

179 

3.342 

206 

3.225 

126 

3.640 

153 

3.473 

180 

3.337 

207 

3.221 

127 

3.633 

154 

3.468 

181 

3.332 

208 

3.217 

128 

3.627 

155 

3.462 

182 

3.328 

209 

3.213 

129 

3.620 

156 

3.457 

183 

3.323 

210 

3.210 

130 

3.614 

157 

3.451 

184 

3.318 

211 

3.206 

131 

3.607 

158 

3.446 

185 

3.314 

212 

3.202 

132 

3.601 

159 

3.440 

186 

3.309 

213 

3.198 

133 

3.594 

160 

3.435 

187 

3.304 

214 

3.195 

134 

3.586 

161 

3.430 

188 

3.300 

215 

3.191 

135 

3.579 

162 

3.424 

189 

3.295 

216 

3.188 

136 

3.573 

163 

3.419 

190 

3.291 

217 

3.184 

137 

3.567 

164 

3.414 

191 

3.287 

218 

3.180 

138 

3.561 

165 

3.409 

192 

3.282 

219 

3.177 

139 

3.555 

166 

3.404 

193 

3.278 

220 

3.173 

140 

3.549 

167 

3.399 

194 

3.274 

221 

3.169 

141 

3.543 

168 

3.394 

195 

3.269 

222 

3.165 

142 

3.537 

169 

3.389 

196 

3.265 

223 

3.162 

143 

3.531 

170 

3.384 

197 

3.261 

224 

3.158 

144 

3.525 

171 

3.380 

198 

3.257 

225 

3.155 

145 

3.519 

172 

3.375 

199 

3.253 

226 

3.151 

146 

3.513 

173 

3.370 

200 

3.249 

227 

3.148 

147 

3.507 

174 

3.365 

201 

3.245 

228 

3.144 

148 

3.501 

175 

3.361 

202 

3.241 

229 

3.141 

149 

3.495 

176 

3.356 

203 

3.237 

230 

3.137 

150 

3.490 

177 

3.351 

204 

3.233 

231 

3.134 

151 

3.484 

178 

3.346 

205 

3.229 

174 


THOMSON  ON  CARNOT'S 


TABLE  II. 

MECHANICAL  EFFECT  IN  FOOT-POUNDS  DUE  TO  A  THER- 
MIC UNIT  CENTIGRADE,  PASSING  FROM  A  BODY,  AT  ANY 
TEMPERATURE  LESS  THAN  230°  TO  A  BODY  AT  0°. 


Superior 
Limit  of 
Temper- 
ature. 

Mechanical 
Effect. 

Superior 
Limit  of 
Temper- 
ature. 

Mechanical 
Effect. 

Superior 
Limit  of 
Temper- 
ature. 

Mechanical 
Effect. 

0 

Ft.-Pouuds. 

<- 

Ft.  -Pounds. 

o 

Ft.-Pounds. 

1 

4.960 

38 

179.287 

75 

337.084 

2 

9.906 

39 

183.761 

76 

341.141 

3 

14.838 

40 

188.223 

77 

345.188 

4 

19.756 

41 

192.673 

78 

->  349.  225 

5 

24.661 

42 

197.111 

79 

353.253 

6 

29.553 

43 

201.537 

80 

357.271 

7 

34.431 

44 

205.951 

81 

361.280 

8 

39.296 

45 

210.353 

82 

365.279 

9 

44.148 

46 

214.743 

83 

369.269 

10 

48.987 

47 

219.121 

84 

373.249 

11 

53.813 

48 

223.487 

85 

377.220 

12 

58.625 

49 

227.842 

86 

381.181 

13 

63.424 

50 

232.185 

87 

385.133 

14 

68.210 

51 

236.516 

88 

389.076 

15 

72.983 

52 

240.835 

89 

393.010 

16 

77.743 

53 

245.143 

90 

396.935 

17 

82.490 

54 

249.439 

91 

400.851 

18 

87.225 

55 

253.724 

92 

404.758 

19 

91.947 

56 

257.997 

93 

408.656 

20 

96.656 

57 

262.259 

94 

412.545 

21 

101.353 

58 

266.509 

95 

416.425 

22 

106.037 

59 

270.748 

96  ; 

420.296 

2& 

110.709 

60 

274.975 

97 

424  159 

24 

115.368 

61 

279.191 

98 

428.013 

25 

120.014 

62 

283.396 

99 

431.858 

26 

124.648 

63 

287.590 

100 

435.695 

27 

129.269 

64 

291.773 

101 

439.524 

28 

133.878 

65 

295.945 

102 

443.344 

29 

138.474 

66 

300.106 

103 

447.156 

30 

143.058 

67 

304.256 

104 

450.960 

31 

147.630 

68 

308.396 

105 

454.756 

32 

152.189 

69 

312.525 

106 

458.544 

33 

156.736 

70 

316.644 

107 

462.324 

34 

161.271 

71 

320.752 

108 

466.096 

35 

165.793 

72 

324.851 

109 

469.860 

36 

170.303 

73 

328.939 

110 

473.617 

37 

174.801 

74 

333.017 

111 

477.366 

MOTIVE  POWER  OF  HEAT, 


175 


TABLE  II.—  (Continued.) 


Superior 
Limit  of 
Temper- 
ature, 

Mechanical 
Effect. 

Superior 
jimit  of 
temper- 
ature. 

Mechanical 
Effect. 

Superior 
Limit  of 
Temper- 
ature. 

Mechanical 
Effect. 

° 

Ft.-Ponnds. 

° 

Ft.-Pounds. 

o 

Ft.-Pounds. 

112 

481.107 

152 

625.105 

192 

760.069 

113 

484.841 

153 

628.578 

193 

763.347 

114 

488.567 

154 

632.046 

194 

766.621 

115 

492.286 

155 

635.508 

195 

769.890 

116 

495.998 

156 

638.965 

196 

773.155 

117 

499.702 

157 

642.416 

197 

776.416 

118 

50*3.399  ! 

158 

645.862 

198 

779.673 

119 

507.088 

159 

649.302 

199 

782.926 

120 

510.770 

160 

652.737 

200 

786.175 

121 

514.445 

161 

656.167 

201 

789.420 

122 

518.113 

162 

659.591 

202 

792.661 

123 

521.174 

163 

663.010 

203 

795.898 

124 

525.428 

164 

666.424 

204 

799.131 

125 

529,075 

165 

669.833 

205 

802.360 

126 

532.715 

166 

673.237 

206 

805.585 

127 

536.348 

167 

676.636 

207 

808.806 

128 

539.975 

168 

680.030 

208 

812.023 

129 

543.595 

169 

683.419 

209 

815.236 

130 

547.209 

170 

686.803 

210 

818.446 

131 

550.816 

171 

690.183 

211 

821.652 

132 

554.417 

172 

693.558 

212 

824.854 

133 

558.051 

173 

696.928 

213 

828.052 

134 

561.597 

174 

700.293 

214 

831.247 

135 

565.176 

175 

703.654 

215 

834.438 

136 

568.749 

176 

707.010 

216 

837.626 

137 

572.316 

177 

710.361 

217 

840.810 

138 

575.877 

178 

713.707 

218 

843.990 

139 

579.432 

179 

717.049 

219 

847.167 

140 

582.981 

180 

720.386 

220 

850.840 

141 

586.524 

181 

723.718 

221 

853.509 

142 

590.061 

182 

727.046 

222 

856.674 

143 

593.592 

183 

730.369 

223 

859.836 

144 

597.117 

184 

733.687 

224 

862.994 

145 

600.636 

185 

737.001 

225 

866.149 

146 

604.099 

186 

740.310 

226 

869.300 

147 

607.656 

187 

743.614 

227 

872.448 

148 

611.157 

188 

746.914 

228 

875.592 

149 

614.652 

189 

750.209 

229 

878.733 

150 

618.142 

190 

753.500 

230 

881.870 

151 

621.626 

191 

756.787 

231 

885.004 

176  THOMSON  ON  CARNOT'S 

Note  on  the  curves  described  in  Clapeyron's 
graphical  method  of  exhibiting  Carnot's  TJieory  of 
the  Steam-Engine. 

39.  At  any  instant  when  the  temperature  of  the 
water  and  vapor  is  t,  during  the  fourth  operation 
(see  above,  §  16,  and  suppose,  for  the  sake  of  sim- 
plicity, that  at  the  beginning  of  the  first  and  at 
the  end  of  the  fourth  operation  the  piston  is  ab- 
solutely in  contact  with  the  surface  of  the  water), 
the  latent  heat  of  the  vapor  must  be  precisely  equal 
to  the  amount  of  heat  that  would  be  necessary  to 
raise  the  temperature  of  the  whole  mass,  if  in  the 
liquid  state,  from  t  to  S.*  Hence,  if  v'  denote  the 
volume  of  the  vapor,  c  the  mean  capacity  for  heat 
of  a  pound  of  water  between  the  temperatures  S 


*  For  at  the  end  of  the  fourth  operation  the  whole  mass 
is  liquid,  and  at  the  temperature  S.  Now,  this  state  might 
be  arrived  at  by  first  compressing  the  vapor  into  water  at 
the  temperature  t,  and  then  raising  the  temperature  of  the 
liquid  to  8 ;  and  however  this  state  may  be  arrived  at,  there . 
cannot,  on  the  whole,  be  any  heat  added  to  or  subtracted 
from  the  contents  of  the  cylinder,  since,  during  the  fourth 
operation,  there  is  neither  gain  nor  loss  of  heat.  This 
reasoning  is,  of  course,  founded  on  Carnot's  fundamental 
principle,  which  is  tacitly  assumed  in  the  commonly-re- 
ceived ideas  connected  with  "Watt's  law,"  the  "latent 
heat  of  steam,"  and  "the  total  heat  of  steam." 


MOTIVE  POWER  OF  HEAT.  177 

and  t,  and  W  the  weight  of  the  entire  mass,,  in 
pounds,  we  have 

kv'=c(S-t)W. 

Again,  the  circumstances  during  the  second  oper- 
ation are  such  that  the  mass  of  liquid  and  vapor 
possesses  H  units  .of  heat  more  than  during  the 
fourth;  and  consequently,  at  the  instant  of  the 
second  operation,  when  the  temperature  is  t,  the 
volume  v  of  the  vapor  will  exceed  v'  by  an  amount 
of  which  the  latent  heat  is  H,  so  that  we  have 

*        '•='+?•     ' 

40.  Now,  at  any  instant,  the  volume  between 
the  piston  and  its  primitive  position  is  less  than 
the  actual  volume  of  vapor  by  the  volume  of  the 
water  evaporated.  Hence,  if  x  and  x'  denote  the 
abscissae  of  the  curve  at  the  instants  of  the  second 
and  fourth  operations  respectively,  when  the  tem- 
perature is  t,  we  have 

x  =  v  —  (TV,    x'=  v'—  vv'  , 
and,  therefore,  by  the  preceding  equations, 

.     .     (a) 
..     (b) 


These  equations,  along  with  y  =  y'  =  p,    .     .     (c) 


178  THOMSON  ON  CARNOT'S 

enable  us  to  calculate,  from  the  data  supplied  by 
Regnault,  the  abscissa  and  ordinate  for  each  of  the 
curves  described  above  (§17)  corresponding  to  any 
assumed  temperature  t.  After  the  explanations  of 
§§  33,  34,  35,  36,  it  is  only  necessary  to  add  that  c 
is  a  quantity  of  which  the  value  is  very  nearly 
unity,  and  would  be  exactly  so  were  the  capacity 
of  water  for  heat  the  same  at  every  temperature 
as  it  is  between  0°  and  1°;  and  that  the  value  of 
c(S  —  t),  for  any  assigned  values  of  S  and  t,  is 
found,  by  subtracting  the  number  corresponding 
to  t  from  the  number  corresponding  to  s,  in  the 
column  headed  "Nombre  des  unites  de  chaleur 
abandonnees  par  un  kilogramme  d'eau  en  descen- 
dant de  T°  a  0°,"  of  the  last  table  (at  the  end  of 
the  tenth  memoir)  of  Kegnault's  work.  By 
giving  S  the  value  230°,  and  by  substituting  suc- 
cessively 220,  210,  200,  etc.,  for  t,  values  for  x,  y, 
x',  y',  have  been  found,  which  are  exhibited  in  the 
table  opposite. 


MOTIVE  POWER  OF  HEAT. 


179 


Tempera- 
tures. 

Volumes  to  be  de- 
scribed by  the  pis- 
ton, to  complete 
the  fourth  opera- 
tion. 

Volumes  from  the 
primitive  position  of 
the  piston  to  those 
occupied  at  instants 
of  the  second  opera- 
tion. 

Pressures  of  sat- 
urated steam,  in 
pounds  on  the 
square  foot. 

t 

xf 

X 

y  =  y'=P 

0° 

1269.  W 

#'4-5.409.5' 

12.832 

10 

639.  6.  W 

z'+2.  847.fi" 

25.567 

20 

337.3.  W 

x'- 

-l.571.fi 

48.514 

30 

185.  5.  W 

x'- 

-  .9062.fi 

88.007 

40 

105.  9.  W 

x'- 

-  .5442.fi 

153.167 

50 

62.62.TF 

x'- 

-  .3392.fi 

256.595 

60 

38.19.TF 

x'- 

-  .2188.fi 

415.070 

70 

21.94.  W 

#'+  .1456.fi 

650.240 

80 

15.38.TF 

x'+  .09962.fi 

989.318 

90 

10.09.T7 

x'+  .06994.fi 

1465.80 

100 

6.744.  W 

x'+  .05026.fi 

2120.11 

110 

4.  578.  IT 

x'+  .03688.fi 

2999.87 

120 

3.141.  TP 

x'-\-  .02758.fi 

4160.10 

130 

2.176.  W 

«'+  .02098.fi 

5663.70 

140 

1.519.  W 

x'+  .01625.fi 

7581.15 

150 

1.058.  W 

x'-\-  .01271.fi 

9990.26 

160 

0.7369.  W 

x'- 

h  .01010.fi 

12976.2 

170 

0.5085.  W 

x'- 

-  .008116.fi 

16630.7 

180 

0.3454.TF 

x'- 

-  .006592.fi 

21051.5 

190 

0.2267.  TF 

x'- 

-  .005406.fi 

26341.5 

200 

0.1409.TF 

X'- 

-  .004472.fi 

32607.7 

210 

0.0784.TF 

x'4-  .003729.fi 

39960.7 

220 

0.3310.TP 

a?'+  .003130.fi 

48512.4 

230 

0 

*'+  .002643.fi 

58376.6 

Appendix. 
(Read  April  30,  1849.) 

41.  In  p.  30  some  conclusions  drawn  by  Carnot 
from  his  general  reasoning  were  noticed ;  accord- 
ing to  which  it  appears,  that  if  the  value  of  //  for 


180  THOMSON  ON  CARNOT'S 

any  temperature  is  known,  certain  information 
may  be  derived  with  reference  to  the  saturated 
vapor  of  any  liquid  whatever,  and,  with  reference 
to  any  gaseous  mass,  without  the  necessity  of  ex- 
perimenting upon  the  specific  medium  considered. 
Nothing  in  the  whole  range  of  Natural  Philosophy 
is  more  remarkable  than  the  establishment  of  gen- 
eral laws  by  such  a  process  of  reasoning.  We  have 
seen,  however,  that  doubt  may  exist  with  reference 
to  the  truth  of  the  axiom  on  which  the  entire  the- 
ory is  founded,  and  it  therefore  becomes  more  than 
a  matter  of  mere  curiosity  to  put  the  inferences 
deduced  from  it  to  the  test  of  experience.  The 
importance  of-  doing  so  was  clearly  appreciated  by 
Carnot ;  and,  with  such  data  as  he  had  from  the 
researches  of  various  experimenters,  he  tried  his 
conclusions.  Some  very  remarkable  propositions 
which  he  derives  from  his  theory  coincide  with 
Dulongand  Petit's  subsequently  discovered  experi- 
mental laws  with  reference  to  the  heat  developed 
by  the  compression  of  a  gas  ;  and  the  experimen- 
tal verification  is  therefore  in  this  case  (so  far  as 
its  accuracy  could  be  depended  upon)  decisive. 
In  other  respects,  the  data  from  experiment  were 
insufficient,  although,  so  far  as  they  were  available 
as  tests,  they  were  confirmatory  of  the  theory. 
42.  The  recent  researches  of  Regnault  add  im- 


MOTIVE  POWER  OF  HEAT.  181 

mensely  to  the  experimental  data  available  for  this 
object,  by  giving  us  the  means  of  determining  with 
considerable  accuracy  the  values  of  //  within  a  very 
wide  range  of  temperature,  and  so  affording  a  trust- 
worthy standard  for  the  comparison  of  isolated 
results  at  different  temperatures,  derived  from  ob- 
servations in  various  branches  of  physical  science. 
In  the  first  section  of  this  Appendix  the  theory 
is  tested,  and  shown  to  be  confirmed  by  the  com- 
parison of  the  values  of  /*  found  above,  with  those 
obtained  by  Carnot  and  Clapeyron  from  the  obser- 
vations of  various  experimenters  on  air,  and  the 
vapors  of  different  liquids.  In  the  second  and 
third  sections  some  striking  confirmations  of  the 
theory  arising  from  observations  by  Dulong,  on 
the  specific  heat  of  gases,  and  from  Mr.  Joule's 
experiments  on  the  heat  developed  by  the  com- 
pression of  air,  are  pointed  out ;  and  in  conclu- 
sion, the  actual  methods  of  obtaining  mechanical 
effect  from  heat  are  briefly  examined  with  refer- 
ence to  their  economy. 

I.  On  the  values  of  ju  derived  by  Carnot  and 
Clapeyron  from  observations  on  Air,  and  on 
the  Vapors  of  various  liquids. 

43.  In    Carnot's  work,    pp.    80-82,   the  mean 
value  of  *  between  0°  and  1°  is  derived  from  the 


182  THOMSON  ON  CARNOT'S 

experiments  of  Delaroche  and  Berard  on  the  spe- 
cific heat  of  gases,  by  a  process  approximately 
equivalent  to  the  calculation  of  the  value  of 

/°/V  for  the  temperature  1°.     There  are  also,  in 
vdq/dv 

the  same  work,  determinations  of  the  values  of  /* 
from  observations  on  the  vapors  of  alcohol  and 
water ;  but  a  table  given  in  M.  Clapeyron's  paper, 
of  the  values  of  ^  derived  from  the  data  supplied 
by  various  experiments  with  reference  to  the  va- 
pors of  ether,  alcohol,  water,  and  oil  of  turpen- 
tine, at  the  respective  boiling-points  of  these 
liquids,  affords  us  the  means  of  comparison  through 
a  more  extensive  range  of  temperature.  In  the 
cases  of  alcohol  and  water,  these  results  ought  of 
course  to  agree  with  those  of  Carnot.  There  are, 
however,  slight  discrepancies  which  must  be  owing 
to  the  uncertainty  of  the  experimental  data.*  In 
the  opposite  table,  Carnot's  results  with  reference 
to  air,  and  Clapeyron's  results  with  reference  to 
the  four  different  liquids,  are  exhibited,  and  com- 
pared with  the  values  of  /*  which  have  been  given 


*  Thus,  from  Carnot's  calculations,  we  find,  in  the  case 
of  alcohol  4.035,  and  in  the  case  of  water  3.648,  instead 
of  3.963  and  3.658,  which  are  Clapeyron's  results  in  the 
same  cases. 


MOTIVE  POWER  OF  HEAT. 


183 


Values  of  M. 

Names  of  the 
Media. 

Temperatures. 

Values  of  /u.. 

deduced 
from  Reg- 
nault's  Ob- 

Differ- 
ences. 

servations. 

0 

(Carnot) 

Air      .  ... 

0.5 

4.377 

4.960 

.383 

Sulphuric 
Ether  

(Boil,  pt.)  35.5 

(Clapeyron) 

4.478 

4.510 

.032 

Alcohol  

78.8 

3.963 

4.030 

.071 

Water  

100 

3.658 

3.837 

.179 

Essence  of 

Turpentine. 

156.8 

3.530 

3.449 

-.081 

above  (Table  I.)  for  the  same  temperatures,  as  de- 
rived from  Regnault's  observations  on  the  vapor 
of  water. 

44.  It  may  be  observed  that  the  discrepancies 
between  the  results  founded  on  the  experimental 
data  supplied  by  the  different  observers  with  ref- 
erence to  water  at  the  boiling-point,  are  greater 
than  those  which  are  presented  between  the  results 
deduced  frotn  any  of  the  other  liquids,  and  water 
at  the  other  temperatures  ;  and  we  may  therefore 
feel  perfectly  confident  that  the  verification  is 
complete  to  the  extent  of  accuracy  of  the  obser- 
vations.* The  considerable  discrepancy  presented 

*  A  still  closer  agreement  must  be  expected  when  more 
accurate  experimental  data  are  afforded  with  reference  to 
the  other  media.  Mons.  Regnault  informs  me  that  he  is 


184  THOMSON  ON  CARNOT'S 

by  Carnot/s  result  deduced  from  experiments  on 
air,  is  not  to  be  wondered  at  when  we  consider  the 
very  uncertain  nature  of  his  data. 

45.  The    fact    of    the    gradual    decrease  of  /* 
through  a  very  extensive   range  of  temperature, 
being  indicated   both  by  Kegnault's    continuous 
series  of  experiments  and  by  the  very  varied  ex- 
periment  on   different    media,    and   in   different 
branches  of  Physical  Science,  must  be  considered 
as  a  striking  verification  of  the  theory. 

II.   On  the  Heat  developed  ~by  the  Compression  of 
Air. 

46.  Let   a   mass   of  air,    occupying  initially  a 
given  volume    V,  under  a  pressure  P,  at  a  tem- 
perature t,  be  compressed  to  a  less   volume    F', 
and  allowed  to  part  with  heat  until  it  sinks  to  its 
primitive  temperature  t.     The  quantity  of  heat 
which  is  evolved  maybe  determined,  according  to 
Carnot's  theory,  when  the  particular  value  of  //, 

engaged  in  completing  some  researches,  from  which  we 
may  expect,  possibly  before  the  end  of  the  present  year, 
to  be  furnished  with  all  the  data  for  five  or  six  different 
liquids  which  we  possess  at  present  for  water.  It  is  there- 
fore to  be  hoped  that,  before  long,  a  most  important  test  of 
the  validity  of  Carnot's  theory  will  be  afforded. 


MOTIVE  POWER  OF  HEAT.  185 

corresponding   to  the  temperature  ty   is  known. 
For,  by  §  30,  equation  (6),  we  have 


where  dq  is  the  quantity  of  heat  absorbed,  when 
the  volume  is  allowed  to  increase  from  v  to  v  +  dv\ 
or  the  quantity  evolved  by  the  reverse  operation. 
Hence  we  deduce 


JH 


(8) 


Now,  °  is  constant,  since  the  temperature 

remains  unchanged  ;  and  therefore  we  may  at 
once  integrate  the  second  number.  By  taking  it 
between  the  limits  V  and  V,  we  thus  find 


.  ,.         . 

where  Q  denotes  the  required  amount  of  heat 
evolved  by  the  compression  from  Vto  P'.  This 
expression  may  be  modified  by  employing  the  equa- 
tions P  V  =  P'  V  =  2).vQ  (1  +  Et)  ;  and  we  thus 
obtain 

EPV  V        EP'V    .       V 

Q  =  °g    7  =  log    -- 


y 

*  The  Napierian  logarithm  of  -~  is  here  understood. 


186  THOMSON  ON  CARNOT'ti 

From  this  result  we  draw  the  following  conclu- 
sion : 

47.  Equal  volumes  of  all  elastic  fluids,  taken  at 
the  same  temperature  and  pressure,  when  com- 
pressed to  smaller  equal  volumes,  disengage  equal 
quantities  of  heat. 

This  extremely  remarkable  theorem  of  Carnot's 
was  independently  laid  down  as  a  probable  experi- 
mental law  by  Dulong,  in  his  "  Recherches  sur  la 
Chaleur  Specifique  des  Flu  ides  Elastiques,"  and  it 
therefore  affords  a  most  powerful  confirmation  of 
the  theory.* 

*  Carnot  varies  the  statement  of  his  theorem,  and  illus- 
trates it  in  a  passage,  pp.  81,  82,  of  which  the  following  is 
translation  : 

"  When  a  gas  varies  in  volume  without  any  change  of  tem- 
perature, the.  quantities  of  heat  absorbed  or  evolved  by  tJiis  gas 
are  in  arithmetical  progression,  if  the  augmentation  or  dimi- 
nutions of  volume  are  in  geometrical  progression. 

"  When  we  compress  a  litre  of  air  maintained  at  the  tem- 
perature 10°,  and  reduce  it  to  half  a  litre,  it  disengages  a 
certain  quantity  of  heat.  If,  again,  the  volume  be  reduced 
from  half  a  litre  to  a  quarter  of  a  litre,  from  a  quarter  to 
an  eighth,  and  so  on  the  quantities  of  heat  successively 
evolved  will  be  the  same. 

"If,  in  place  of  compressing  the  air,  we  allow  it  to  ex- 
pand to  two  litres,  four  litres,  eight  litres,  etc.,  it  will  be 
necessary  to  supply  equal  quantities  of  heat  to  maintain  the 
temperature  always  at  the  same  degree. " 


MOTIVE  POWER  OF  HEAT.  187 

48.  In  some  very  remarkable  researches  made 
by  Mr.  Joule  upon  the   heat   developed   by  the 
compression  of  air,  the  quantity  of  heat  produced 
in  different  experiments  has  been  ascertained  with 
reference  to   the  amount   of  work   spent  in   the 
operation.     To  compare  the  results  which  he  has 
obtained  with  the  indications  of  theory,  let  us  de- 
termine the  amount  of  work  necessary  actually  to 
produce  the  compression  considered  above. 

49.  In  the  first  place,  to  compress  the  gas  from 
the  volume  v  +  dv  to  v,  the  work  required  is  pdv, 
or,  since 

PV  =JV>.(1  +  Et), 


Hence,  if  we  denote  by  W  the  total  amount  of 
work  necessary  to  produce  the  compression  from 
Fto  V9  we  obtain,  by  integration, 

W  =  pQVQ(I  +  fit)  log  y,. 

Comparing  this  with  the  expression  above,  we  find 
W_»(1+M)  .    . 

-Q  =      —JE~ 
50.  Hence  we  infer  that  — 
(1)   The  amount  of  work  necessary  to  produce 
a  unit  of  heat  by  the  compression  of  a  gas  is  the 
same  for  all  gases  at  the  same  temperature; 


188 


THOMSON  ON  CARNOT'S 


(2)  And  that  the  quantity  of  heat  evolved  in 
all  circumstances,  when  the  temperature  of  the 
gas  is  given,  is  proportional  to  the  amount  of  work 
spent  in  the  compression. 

51.  The  expression  for  the  amount  of  work  nec- 
essary to  produce  a  unit  of  heat  is 

Ml  +  Et) 

E 

and  therefore  Kegnault's  experiments  on  steam 
are  available  to  enable  us  to  calculate  its  value  for 
any  temperature.  By  finding  the  values  of  yu  at 
0°,  10°,  20°,  etc.,  from  Table  I.,  and  by  substi- 
tuting successively  the  values  0,  10,  20,  etc.,  for  f, 
the  following  results  have  been  obtained : 
TABLE  OF  THE  VALUES  OF 


Work  requisite  to 
produce  a  unit 
of  Heat  by  the 
compression  of 

Temperature 
of  the  Gas. 

Work  requisite  to 
produce  a  unit 
of  Heat  by  the 
compression  of 

Temperature 
of  the  Gas. 

a  Gas. 

a  Gas. 

Ft.  -pounds. 

0 

Ft.-pounds. 

o 

1357.1 

0 

1446.4 

120 

1368.7 

10 

1455.8 

130 

1379.0 

20 

1465.3 

140 

1388.0 

30 

1475.8 

150 

1395.7 

40 

1489.2 

160 

1401.8 

50 

1499.0 

170 

1406.7 

60 

1511.3 

180 

1412.0 

70 

1523.5 

190 

1417.6 

80 

1536.5 

200 

1424.0 

90 

1550.2 

210 

1430.6 

100 

1564.0 

220 

1438.2 

110 

1577.8 

230 

MOTIVE  POWER  OF  HEAT.  189 

Mr.  Joule's  experiments  were  all  conducted  at 
temperatures  from  50°  to  about  GO0  Fahr.,  or  from 
10°  to  16°  Cent.;  and  consequently,  although  some 
irregular  differences  in  the  results,  attributable  to 
errors  of  observation  inseparable  from  experiments 
of  such  a  very  difficult  nature,  are  presented, 
no  regular  dependence  on  the  temperature  is  ob- 
servable. From  three  separate  series  of  experi- 
ments, Mr.  Joule  deduces  the  following  numbers 
for  the  work,  in  foot-pounds,  necessary  to  produce 
a  thermic  unit  Fahrenheit  by  the  compression  of 
a  gas. 

820,  814,  760. 

Multiplying  these  by  1.8,  to  get  the  corresponding 
number  for  a  thermic  unit  Centigrade,  we  find 

1476,  1465,  and  1368. 

The  largest  of  these  numbers  is  most  nearly 
conformable  with  Mr.  Joule's  views  of  the  relation 
between  such  experimental  '''equivalents,"  and 
others  which  he  obtained  in  his  electro-magnetic 
researches  ;  but  the  smallest  agrees  almost  perfect- 
ly with  the  indications  of  Carnot's  theory ;  from 
which,  as  exhibited  in  the  preceding  table,  we 
should  expect,  from  the  temperature  in  Mrc  Joule's 
experiments,  to  find  a  number  between  1369  and 
1379  as  the  result.* 

*  The  best  figure  (1896)  is  J  -  778  ft.-lbs.  =  1  B.T.U.,  or 
J  =  426.8  kgin.  =  1  calorie,  aud  probably  with  great  ac- 
curacy. 


190  '    THOMSON  ON  CARNOT'S 

III.   On  the  Specific  Heats  of  Gases. 

52.  The  following    proposition    is   proved  by 
Carnot  as  a  deduction  from  his  general  theorem 
regarding  the  specific  heats  of  gases. 

The  excess  of  specific  heat*  under  a  constant 
pressure  above  the  specific  heat  at  a  constant  volume, 
is  the  same  for  all  gases  at  the  same  temperature 
and  pressure. 

53.  To  prove  this  proposition,  and  to  determine 
an  expression  for  the  "excess"  mentioned  in  its 
enunciation,  let  us  suppose  a  unit  of  volume  of  a 
gas  to   be    elevated   in   temperature   by   a   small 
amount,  T.     The  quantity  of  heat  required  to  do 
this  will  be  AT,  if  A  denote  the  specific  heat  at  a 
constant  volume.     Let  us  next  allow  the  gas  to 

O 

expand  without  going  down  in  temperature,  until 
its  pressure  becomes  reduced  to  its  primitive  value. 

ET 

The  expansion  which  will  take  place  will  be  --  , 

1  -j-  Et 

if  the  temperature  be  denoted  by  t  ;  and  hence, 
by  (8),  the  quantity  of  heat  that  must  be  supplied, 
to  prevent  any  lowering  of  temperature,  will  be 


Et 


*  Or  the  capacity  of  a  unit  of  volume  for  heat. 


MOTIVE  POWER  OF  HEAT.  191 

Hence  the  total  quantity  added  is  equal  to 


But,  since  B  denotes  the  specific  heat  under  con- 
stant pressure,  the  quantity  of  heat  requisite  to 
bring  the  gas  into  this  state,  from  its  primitive 
condition,  is  equal  to  Br\  and  hence  we  have 


IV.   Comparison  of  the  Relative  Advantages  of  the 
Air-engine  and  Steam-engine. 

54.  In  the  use  of  water-wheels  for  motive  power, 
the  economy  of  the  engine  depends  not  only  upon 
the  excellence  of  its  adaptation  for  actually  trans- 
mitting any  given  quantity  of  water  through  it, 
and  producing  the  equivalent  of  work,  but  upon 
turning  to  account  the  entire  available  fall;  so,  as 
we  are  taught  by  Carnot,  the  object  of  a  thermo- 
dynamic  engine  is  to  economize  in  the  best  possible 
way  the  transference  of  all  the  heat  evolved,  from 
bodies  at  the  temperature  of  the  source,  to  bodies 
at  the  lowest  temperature  at  which  the  herat  can  be 
discharged.  With  reference,  then,  to  any  engine  of 
the  kind,  there  will  be  two  points  to  be  considered: 

(1)  The  extent  of  ihefall  utilized. 


192  THOMSON   ON  CARNOT'8 

(2)  The  economy  of  the  engine,  with  the  fall 
which  it  actually  uses. 

55.  In  the  first  respect,  the  air-engine,  as  Carnot 
himself  points  out,  has  a  vast  advantage  over  the 
steam-engine;  since  the  temperature  of  the  hot 
part  of  the  machine  may  be  made  very  much 
higher  in  the  air-engine  than  would  be  possible  in 
the  steam-engine,  on  account  of  the  very  high 
pressure  produced  in  the  boiler,  by  elevating  the 
temperature  of  the  water  which  it  contains  to  any 
considerable  extent  above  the  atmospheric  boiling- 
point.  On  this  account  a  "perfect  air-engine " 
would  be  a  much  more  valuable  instrument  than  a 
"  perfect  steam-engine."  * 

*  Carnot  suggests  a  combination  of  the  two  principles, 
with  air  as  the  medium  for  receiving  the  heat  at  a  very 
high  temperature  from  the  furnace;  ^nd  a  second  medium, 
alternately  in  the  state  of  saturated  vapor  and  liquid  water, 
to  receive  the  heat,  discharged  at  aii  intermediate  temper- 
ature from  the  air,  and  transmit  it  to  the  coldest  part  of 
the  apparatus.  It  is  possible  that  a  complex  arrangement 
of  this  kind  might  be  invented  which  would  enable  us  to 
take  the  heat  at  a  higher  temperature,  and  discharge  it  at  a 
lower  temperature  than  would  be  practicable  in  any  simple 
air-engine  or  simple  steam-engine.  If  so,  it  would  no 
doubt  be  equally  possible,  and  perhaps  more  convenient, 
to  employ  steam  alone,  but  to  use  it  at  a  very  high  tem- 
perature not  in  contact  with  water  in  the  hottest  part  of 


MOTIVE  POWER  OF  HEAT.  193 

Neither  steam-engines  nor  air-engines,  however, 
are  nearly  perfect;  and  we  do  not  know  in  which 
of  the  two  kinds  of  machine  the  nearest  approach 
to  perfection  may  be  actually  attained.  The  beau- 
tiful engine  invented  by  Mr.  Stirling  of  Galston 
may  be  considered  as  an  excellent  beginning  for 
the  air-engine;*  and  it  is  only  necessary  to  com- 
pare this  with  Newcomen's  steam-engine,  and  con- 
sider what  Watt  has  effected,  to  give  rise  to  the 
most  sanguine  anticipations  of  improvement. 

V.  On  the  Economy  of  Actual  Steam-engines. 

56.  The  steam-engine  being  universally  em- 
ployed at  present  as  the  means  for  deriving  motive 
power  from  heat,  it  is  extremely  interesting  to 
examine,  according  to  Carnot's  theory,  the  econ- 
omy actually  attained  in  its  use.  In  the  first 


the  apparatus,  instead  of,  as  in  the  steam-engine,  always 
in  a  saturated  state. 

*  It  is  probably  this  invention  to  which  Carnot  alludes 
in  the  following  passage:  "II  a  ete  fait,  dit-on,  tout  re- 
cemment  en  Angleterre  des  essais  heureujt  sur  le  de- 
veloppement  de  la  puissance  motrice  par  1'action  de  la 
chaleur  sur  1'air  atmospherique.  Nous  ignorons  entiere- 
ment  ne  quoi  ces  essais  ont  consiste,  si  toutefois  ils  sont 
reels." 


194  THOMSON  ON  CARNOT'S 

place  we  remark,  that  out  of  the  entire  "fall" 
from  the  temperature  of  the  coals  to  that  of  the 
atmosphere  it  is  only  part — that  from  the  tem- 
perature of  the  boiler  to  the  temperature  of  the 
condenser — that  is  made  available;  while  the  very 
great  fall  from  the  temperature  of  the  burning 
coals  to  that  of  the  boiler,  and  the  comparatively 
small  fall  from  the  temperature  of  the  condenser 
to  that  of  the  atmosphere,  are  entirely  lost  as 
far  as  regards  the  mechanical  effect  which  it  is 
desired  to  obtain.  We  infer  from  this,  that  the 
temperature  of  the  boiler  ought  to  be  kept  as 
high  as,  according  to  the  strength,  is  consistent 
with  safety,  while  that  of  the  condenser  ought 
to  be  kept  as  nearly  down  at  the  atmospheric 
temperature  as  possible.  To  take  the  entire  ben- 
efit of  the  actual  fall,  Carnot  showed  that  the 
"  principle  of  expansion"  must  be  pushed  to  the 
utmost.* 

*  From  this  point  of  view,  we  see  very  clearly  how  im- 
perfect is  the  steam-engine,  even  after  all  Watt's  improve- 
ments. For  to  "  push  the  principle  of  expansion  to  the 
utmost, "  we  must  allow  the  steam,  before  leaving  the  cyl- 
inder, to  expand  until  its  pressure  is  the  same  as  that  of 
the  vapor  in  the  condenser.  According  to  "Watt's  law/5 
its  temperature  would  then  be  the  same  as  (actually  a  little 
above,  as  Regnault  has  shown)  that  of  the  condenser,  and 


MOTIVE  POWER  OF  HEAT.  195 

57.  To  obtain  some  notion  of  the  economy  which 
has  actually  been  obtained,  we  may  take  the  al- 
leged performances  of  the  best  Cornish  engines, 
aud  some  other  interesting  practical  cases,  as  ex- 
amples.* 

(1)  The  engine  of  the  Fowey  Consols  mine  was 
reported,  in  1845,  to  have  given  125,089,000  foot- 
pounds of  effect,  for  the  consumption  of  one 
bushel  or  94  Ibs.  of  coals.  Now  the  average  amount 
evaporated  from  Cornish  boilers,  by  one  pound  of 
coal,  is  8£  Ibs.  of  steam  ;  and  hence  for  each 
pound  of  steam  evaporated  156,556  foot-pounds  of 
work  are  produced. 

The  pressure  of  the  saturated  steam  in  the  boiler 
may  be  taken  as  3J  atmospheres;!  and,  conse- 


hence  the  steam-engine  worked  in  this  most  advantageous 
way  has  in  reality  the  very  fault  that  Watt  found  in  New- 
comen's  engine.  This  defect  is  partially  remedied  by 
Hornblower's  system  of  using  a  separate  expansion  cylin- 
der, an  arrangement  the  advantages  of  which  did  not 
escape  Caruot's  notice,  although  they  have  not  been  recog- 
nized extensively  among  practical  engineers,  until  within 
the  last  few  years. 

*  I  am  indebted  to  the  kindness  of  Professor  Gordon  of 
Glasgow  for  the  information  regarding  the  various  cases 
given  in  the  text. 

f  In  different  Cornish  engines,  the  pressure  in  the  boiler 


196  THOMSON  ON  CARNOT'S 

quently,  the  temperature  of  the  water  will  be  140°. 
Now  (Regnault,  end  of  Memoire  X.)  the  latent 
heat  of  a  pound  of  saturated  steam  at  140°  is  508, 
and  since,  to  compensate  for  each  pound  of  steam 
removed  from  the  boiler  in  the  working  of  the 
engine,  a  pound  of  water,  at  the  temperature  of 
the  condenser,  which  may  be  estimated  at  30°,  is 
introduced  from  the  hot- well;  it  follows  that  618 
units  of  heat  are  introduced  to  the  boiler  for  each 
pound  of  water  evaporated.  But  the  work  pro- 
duced, for  each  pound  of  water  evaporated,  was 
found  above  to  be  156,556  foot-pounds.  Hence 
JJ5AV~>  or  253  foot-pounds,  is  the  amount  of  work 
produced  for  each  unit  of  heat  transmitted  through 
the  Fowey  Consols  engine.  Now  in  Table  II.  we 
find  583.0  as  the  theoretical  effect  due  to  a  unit  de- 
scending from  140°  to  0°,  and  143  as  the  effect  due 
to  a  unit  descending  from  30°  to  0°.  The  difference 
of  these  numbers,  or  440,*  is  the  number  of  foot- 
is  from  2|  to  5  atmospheres;  and,  therefore,  as  we  find 
from  Regnault's  table  of  the  pressure  of  saturated  steam, 
the  temperature  of  the  water  in  the  boiler  must,  in  all  of 
them,  lie  between  128°  and  152°.  For  the  better  class  of 
engines,  the  average  temperature  of  the  water  in  the  boiler 
may  be  estimated  at  140°,  the  corresponding  pressure  of 
steam  being  3|  atmospheres. 
*  This  number  agrees  very  closely  with  the  number 


MOTIVE  PO  WEE  OF  HEAT.  197 

pounds  of  work  that  &  per  feet  engine  with  its  boiler 
at  140°  and  its  condenser  at  30°  would  produce  for 
each  unit  of  heat  transmitted.  Hence  the  Fowey 
Consols  engine,  during  the  experiments  reported 
on,  performed  f  JJ  of  its  theoretical  duty,  or  57J- 
per  cent. 

(2)  The  best  duty  on  record,  as  performed  by  an 
engine  at  work  (not  for  merely  experimental  pur- 
poses), is  that  of  Taylor's  engine,  at  the  United 
Mines,   which    in  1840  worked  regularly   for  sev- 
eral months  at  the  rate  of  98,000,000  foot-pounds 
for  each  bushel  of  coals  burned.     This  is  ^/j,  or 
.784  of  the  experimental  duty  reported  in  the  case 
of    the  Fowey   Consols  engine.     Hence  the  best 
useful  work  on  record  is  at  the  rate  of  198.3  foot- 
pounds for  each  unit  of  heat  transmitted,  and  is 
-Y^3,  or  45  per  cent  of  the  theoretical  duty,  on 
the  supposition  that  the  boiler  is  at  140°  and  the 
condenser  at  30°. 

(3)  French  engineers  contract  (in  Lille,  in  1847, 
for  example)  to  make  engines  for  mill-power  which 
will  produce  30,000  metre-pounds  or  98,427  foot- 
pounds of  work  for  each  pound  of  steam  used.     If 

corresponding  to  the  fall  from  100°  to  0°,  given  in  Table 
II.  Hence,  tlie  fall  from  140°  to  30°  of  the  scale  of  the 
air-thermometer  is  equivalent,  with  reference  to  motive 
power,  to  the  fall  from  100°  to  0°. 


198  THOMSON  ON  CAENOT'S 

we  divide  this  by  618,  we  find  159  foot-pounds  for 
the  work  produced  by  each  unit  of  heat.  This  is 
36.1  per  cent  of  440,  the  theoretical  duty.* 

(4)  English  engineers  have  contracted  to  make 
engines  and  boilers  which  will  require  only  3J  Ibs. 
of  the  best  coal  per  horse-power  per  hour.  Hence 
in  such  engines  each  pound  of  coal  ought  to  pro- 
duce 565,700  foot-pounds  of  work,  and  if  7  Ibs.  of 
water  be  evaporated  by  each  pound  of  coal,  there 
would  result  83,814  foot-pounds  of  work  for  each 
pound  of  water  evaporated.  If  the  pressure  in  the 

*  It  being  assumed  that  the  temperatures  of  the  boiler 
and  condenser  are  the  same  as  those  of  the  Cornish  en- 
gines. If,  however,  the  pressure  be  lower,  two  atmos- 
pheres, for  instance,  the  numbers  would  stand  thus:  The 
temperature  in  the  boiler  would  be  only  121.  Conse- 
quently, for  each  pound  of  steam  evaporated,  only  614 
units  of  heat  would  be  required  ;  and  therefore  the  work 
performed  for  each  unit  of  heat  transmitted  would  be 
160.3  foot-pounds,  which  is  more  than  according  to  the 
estimate  in  the  text.  On  the  other  hand,  the  range  of  tem- 
peratures, or  the  fall  utilized,  is  only  from  131  to  30,  in- 
stead of  from  140  to  30°,  and,  consequently  (Table  II.),  the 
theoretical  duty  for  each  unit  of  heat  is  only  371  foot- 
pounds. Hence,  if  the  engine,  to  work  according  to  the 
specification,  requires  a  pressure  of  only  15  Ibs.  on  the 
square  inch  (i.e.,  a  total  steam-pressure  of  two  atmos- 
pheres), its  performance  is  -l  $%'•£,  or  43.2  per  cent  of  its 
theoretical  duty. 


MOTIVE  POWER  OF  HEAT.  199 

boiler  be  3^  atmospheres  (temperature  140°)  the 
amount  of  work  for  each  unit  of  heat  will  be 
found,  by  dividing  this  by  618,  to  be  130.7  foot- 
pounds, which  is  -VA5  or  29.7  per  cent  of  the  theo- 
retical duty.* 

(5)  The  'actual  average  of  work  performed  by 
good  Cornish  engines   and  boilers   is   55,000,000 
foot-pounds  for  each  bushel  of  coal,  or  less  than 
half  the  experimental  performance  of  the  Fowey 
Consols  engine,  more  than   half  the   actual   duty 
performed  by  the  United  Mines  engine  in  1840; 
in  fact,  about  25  per  cent  of  the  theoretical  duty. 

(6)  The  average  performances  of  a  number   of 
Lancashire  engines  and  boilers  have  been  recently 
found  to  be   such  as  to  require  12  Ibs.  of   Lanca- 
shire coal  per  horse-power  per  hour  (i.e.,  for  per- 
forming 60  X  33,000  foot-pounds),  and  of  a  num- 
ber of  Glasgow  engines  such  as  to  require  15  Ibs. 
(of  a  somewhat  inferior  coal)  for  the  same  effect. 
There  are,  however,  more  than  twenty  large  en- 
gines in  Glasgow  at  presentf  which  work   with  a 

*  If,  in  this  case  again ,  the  pressure  required  in  the  boiler 
to  make  the  engine  work  according  to  the  contract  were 
only  15  Ibs.  on  the  square  inch,  we  should  have  a  different 
estimate  of  the  economy,  for  which  see  Table  B,  at  the 
end  of  this  paper. 

f  These  engines  are  provided  with  separate  expansion 


200  THOMSON  ON  CARNOT'S 

consumption  of  only  6J  Ibs.  of  dross,  equivalent 
to  5  Ibs.  of  the  best  Scotch  or  4  Ibs.  of  the  best 
Welsh  coal,  per  horse-power  per  hour.  The 
economy  may  be  estimated  from  these  data,  as  in 
the  other  cases,  on  the  assumption  which,  with 
reference  to  these,  is  the  most  probable  we  can 
make,  that  the  evaporation  produced  by  a  pound 
of  best  coal  is  7  Ibs.  of  steam. 

58.  The  following  tables  afford  a  synoptic  view 
of  the  performances  and  theoretical  duties  in  the 
various  cases  discussed  above. 

In  Table  A  the  numbers  in  the  second  column 
are  found  by  dividing  the  numbers  in  the  first  by 
8J  in  cases  (1),  (2),  and  (5),  and  by  7  in  cases  (4), 
(6),  and  (7),  the  estimated  numbers  of  pounds  of 
steam  actually  produced  in  the  different  boilers  by 
the  burning  of  1  Ib.  of  coal. 

The  numbers  in  the  third  column  are  found 
from  those  in  the  second,  by  dividing  by  618  in 
Table  A,  and  614  in  Table  B,  which  are  respec- 
tively the  quantities  of  heat  required  to  convert  a 
pound  of  water  taken  from  the  hot-well  at  30°, 
into  saturated  steam,  in  the  boiler,  at  140°  or  at 
121°. 


cylinders,  which  have  been  recently  added  to  them  by 
Mr.  M 'Naught  of  Glasgow. 


MOTIVE  POWER  OF  HEAT.  201 

With  reference  to  the  cases  (3),  (4),  (6),  (7),  the 
hypothesis  of  Table  B  is  probably  in  general  nearer 
the  truth  than  that  of  Table  A.  In  (4),  (6),  and 
(7),  especially  upon  hypothesis  B,  there  is  much 
uncertainty  as  to  the  amount  of  evaporation  that 
will  be  actually  produced  by  1  Ib.  of  fuel.  The 
assumption  on  which  the  numbers  in  the  second 
column  in  Table  B  are  calculated,  is,  that  each 
pound  of  coal  will  send  the  same  number  of  units 
of  heat  into  the  boiler,  whether  hypothesis  A  or 
hypothesis  B  be  followed.  Hence,  except  in  the 
case  of  the  French  contract,  in  which  the  evapora- 
tion, not  the  fuel,  is  specified,  the  numbers  in  the 
third  column  are  the  same  as  those  in  the  third 
column  of  Table  A. 


202 


THOMSON  ON  CARNOT'S 


TABLE  A. 

VARIOUS  ENGINES  IN  WHICH  THE  TEMPERATURE  OP  THE 
BOILER  is  140°  C.  AND  THAT  OF  THE  CONDENSER  30°  C. 

Tfworetical  Duty  for  each  Unit  of  Heat  transmitted,  440* 
foot-pounds. 


CASES. 

Work  pro- 
duced for 
each  Ib.  of 

Work  pro- 
duced for 
each  Ib.  of 

Work  pro- 
duced for 
each   unit 

Percent- 
age    of 
theo- 

coal   con- 

watereva- 

of       heat 

retical 

sumed. 

porated. 

transmit- 
ted. 

duty. 

Ft.-lbs. 

Ft.-lbs. 

Ft.-lbs. 

(1)  Fowey  Consols  experi-  ) 
ment,  reported  in  1845  j 

1,330,734 

156,556 

253 

57.5 

(2)  Taylor's  engine  at  the  i 
United   Mines,  work-  v 
ing  in  1840                      f 

1,042,553 

122,653 

198.4 

45.1 

(3)  French  engines,  accord-  ) 
ing  to  contract  f 

98,427 

159 

36.1 

(4)  English     engines,     ac-  ( 
cording  to  contract.  .  f 

565,700 

80,814 

130.8 

29.7 

(5)  Average    actual     per-  i 

formance  of  Cornish  V 

585,106 

68,836 

111.3 

25.3 

engines  ) 

(6)  Common  engines,  con-~| 
suming  12  IDS.  of  best  ', 
coal  per  horse-power  [ 

165,000 

23,571 

38.1 

8.6 

per  hour  J 

(7)  Improved  engines  withl 
expansion    cylinders, 

consuming  an  equiva-  1 
lent  to  4  Ibs.  of  best  f 

495,000 

70,710 

114.4 

26 

coal  per  horse-power 

per  hour  

*  [Note  added  March  15,  1881.    Total  work  for  thermal  unit,  1390 
(Joule),  377.1  corrected  by  the  dynamical  theory,  March  15, 1851. 
377.1=  .2713X1390, 
853  =  .1820  X  1390  =          X  1390.] 


MOTIVE  POWER  OF  HEAT. 


203 


TABLE  B. 

VARIOUS  ENGINES  IN  WHICH  THE  TEMPERATURE  OF  THE 
BOILER  is  121°  C.*  AND  THAT  OP  THE  CONDENSER  30°  C. 

Theoretical  Duty  for  each  Unit  of  Heat  transmitted,  371 
foot-pounds. 


CASES. 

Work  pro- 
duced for 
each  Ib.  of 
coal    con- 
sumed. 

Work     pro- 
duced   for 
each  Ib.  of 
water  eva- 
porated. 

Work  pro- 
duced for 
each   unit 
of       heat 
transmit- 
ted. 

Per- 
cent- 
age of 
theo- 
retical 
duty. 

Ft.  -Ibs. 

Ft.-lbs. 

Ft.-lbs. 

(3)  French  engines,  accord- 

98,427 

160.3 

43.2 

ing  to  contract  

(4)  English     engines,    ac- 
cording to  contract.. 

565,700 

fit  x  80,814 

130.8 

35 

(6)  Common  engines,  con- 

suming 12  Ibs.  of  coal 
per  horse-power   per 
hour   

165,000 

fit  x  23,  571 

38.1 

10.3 

(7)  Improved  engines  with 

expansion    cylinders, 
consuming  an  equiva- 
lent to  4  Ibs.  best  coal 

495,000 

iif  x  70,710 

114.4 

30.7 

per  horse-power  per 
hour 

*  Pressure  15  Ibs.  on  the  square  inch. 


APPENDIX  A. 


EXTRACTS  FROM  UNPUBLISHED  WRITINGS 
OF  CARNOT. 

I.  NOTES. 

LET  us  first  open  at  the  memoranda  relating  to 
his  daily  occupations : 

"Plan  in  the  morning  the  work  of  the  day,  and 
reflect  in  the  evening  on  what  has  been  done." 

"  Carry  when  walking  a  book,  and  a  note-book 
to  preserve  the  ideas,  and  a  piece  of  bread  in  order 
to  prolong  the  walk  if  need  be." 

"Vary  the  mental  and  bodily  exercises  with 
dancing,  horsemanship,  swimming,  fencing  with 
sword  and  with  sabre,  shooting  with  gun  and  pistol, 
skating,  the  sling,  stilts,  tennis,  bowls;  hop  on  one 
foot,  cross  the  arms,  jump  high  and  far,  turn  on 
one  foot  propped  against  the  wall,  exercise  in  shirt 
in  the  evening  to  get  up  a  perspiration  before  going 
to  bed ;  turning,  joinery,  gardening,  reading  while 
walking,  declamation,  singing,  violin,  versification, 
musical  composition ;  eight  hours  of  sleep  ;  a  walk 
on  awakening,  before  and  after  eating  ;  great  so- 

205 


206  APPENDIX  A. 

briety ;  eat  slowly,  little,  and  often ;  avoid  idle- 
ness and  useless  meditation/' 

Then  come  more  general  precepts  : 

"  Adopt  good  habits  when  I  change  my  method 
of  life." 

"Never  turn  to  the  past  unless  to  enlighten  the 
future.  Regrets  are  useless/' 

"Form  resolutions  in  advance  in  order  not 
to  reflect  during  action.  Then  obey  thyself 
blindly." 

"The  promptitude  of  resolutions  most  fre- 
quently accords  with  their  justice/' 

"  Yield  frequently  to  the  first  inspiration.  Too 
much  meditation  on  the  same  subject  ends  by  sug- 
gesting the  worst  part,  or  at  least  causes  loss  of 
precious  time." 

"Suffer  slight  disagreeables  without  seeming  to 
perceive  them,  but  repulse  decisively  any  one  who 
evidently  intends  to  injure  or  humiliate  you." 

"  One  should  never  feign  a  character  that  he 
has  not,  or  affect  a  character  that  he  cannot  sus- 
tain." 

"  Self-possession  without  self-sufficiency.  Cour- 
age without  effrontery." 

"  Make  intimate  acquaintances  only  with  much 
circumspection ;  perfect  confidence  in  those  who 


APPENDIX  A.  207 

have  been  thoroughly  tested.  Nothing  to  do  with 
others/' 

"  Question  thyself  to  learn  what  will  please 
others/' 

"  No  useless  discourse.  All  conversation  which 
does  not  serve  to  enlighten  ourselves  or  others, 
to  interest  the  heart  or  amuse  the  mind,  is  hurt- 
ful." 

"  Speak  little  of  what  you  know,  and  not  at  all 
of  what  you  do  not  know." 

"Why  not  say  more  frequently,  'I  do  not 
know'?" 

"  Speak  to  every  one  of  that  which  he  knows 
best.  This  will  put  him  at  his  ease,  and  be  profit- 
able to  you/' 

"Abstain  from  all  pleasantry  which  could 
wound." 

"  Employ  only  expressions  of  the  most  perfect 
propriety. " 

"  Listen  attentively  to  your  interlocutor,  and  so 
prepare  him  to  listen  in  the  same  way  to  your  reply, 
and  predispose  him  in  favor  of  your  arguments." 

"  Show  neither  passion  nor  weariness  in  discus- 
sion." 

' '  Never  direct  an  argument  against  any  one.  If 
you  know  some  particulars  against  your  adversary, 
you  have  a  right  to  make  him  aware  of  it  to  keep 


208  APPENDIX  A. 

him  under  control,  but  proceed  with  discretion, 
and  do  not  wound  him  before  others." 

"  When  discussion  degenerates  into  dispute,  be 
silent;  this  is  not  to  declare  yourself  beaten." 

"  How  much  modesty  adds  to  merit !  A  man  of 
talent  who  conceals  his  knowledge  is  like  a  branch 
bending  under  a  weight  of  fruit." 

"Why  try  to  be  witty?  I  would  rather  be 
thought  stupid  and  modest  than  witty  and  pre- 
tentious." 

"  Men  desire  nothing  so  much  as  to  make  them- 
selves envied." 

' '  Egotism  is  the  most  common  and  most  hated 
of  all  vices.  Properly  speaking,  it  is  the  only  one 
which  should  be  hated." 

"  The  pleasures  of  self-love  are  the  only  ones 
that  can  really  be  turned  into  ridicule. " 

"I  do  not  know  why  these  two  expressions, 
good  sense  and  common  sense,  are  confounded. 
There  is  nothing  less  common  than  good  sense." 

' '  The  strain  of  suffering  causes  the  mind  to 
decay." 

We  will  quote  one  of  those  misanthropic  sallies 
the  rarity  of  which  we  are  glad  to  remark  : 

"  It  must  be  that  all  honest  people  are  in  the 
galleys;  only  knaves  are  to  be  met  with  elsewhere." 


APPENDIX  A.  209 

But  serenity  of  mind  returns  immediately  after 
the  above  : 

"  I  rejoice  for  all  the  misfortunes  which  might 
have  happened  to  me,  and  which  I  have  escaped." 

' i  Life  is  a  short  enough  passage.  I  am  half  the 
journey.  I  will  complete  the  remainder  as  I  can." 

"Hope  being  the  greatest  of  all  blessings,  it  is 
necessary,  in  order  to  be  happy,  to  sacrifice  the 
present  to  the  future." 

"  Let  us  not  be  exacting;  perfection  is  so  rare/' 

"  Indulgence  !     Indulgence  !" 

"The  more  nearly  an  object  approaches  perfec- 
tion, the  more  we  notice  its  slightest  defects." 

"To  neglect  the  opportunity  of  an  innocent 
pleasure  is  a  loss  to  ourselves.  It  is  to  act  like  a 
spendthrift." 

"Recherche  pleasures  cause  simple  pleasures  to 
lose  all  their  attractions." 

"It  may  sometimes  be  necessary  to  yield  the 
right,  but  how  is  one  to  recover  it  when  wanted  ?" 

"Love  is  almost  the  only  passion  that  the  good 
man  may  avow.  It  is  the  only  one  which  accords 
with  delicacy." 

"Do  nothing  that  all  the  world  may  not  know." 

"  The  truly  wise  man  is  he  who  loves  virtue  for 
its  own  sake." 


210  APPENDIX  A. 

"  We  say  that  man  is  an  egotist,  and  neverthe- 
less his  sweetest  pleasures  come  to  him  through 
others.  He  only  tastes  them  on  condition  of  shar- 
ing them." 

"  If  one  could  continually  satisfy  his  desires,  he 
would  never  have  time  to  desire.  Happiness  then 
is  necessarily  composed  of  alternatives.  It  could 
not  exist  at*a  constant  level." 

On  the  subject  of  nations  and  conquerors  : 

"  To  each  conqueror  can  be  said,  when  he  has 
ceased  tormenting  our  poor  globe,  '  Would  you 
not  have  been  able  to  tilt  equally  well  against  a 
little  globe  of  pasteboard  ? ' ' 

"  The  laws  of  war,  do  they  say  ?  As  if  war 
were  not  the  destruction  of  all  laws." 

"  War  has  been  represented  as  necessary  to  pre- 
vent the  too  rapid  increase  of  the  population,  but 
war  mows  down  the  flower  of  the  young  men, 
while  it  spares  the  men  disgraced  by  nature. 
Hence  it  tends  to  the  degeneration  of  the  species." 

Then  the  writer  turns  his  shafts  against  medi- 
cine : 

"  In  some  respects  medicine  is  directly  opposed 

•  to  the  will  of  nature,  which  tends  to  perpetuate  the 

strongest  and  best  of  the  species,  and  to  abandon 


APPENDIX  A.  211 

the  delicate  to  a  thousand  forms  of  destruction. 
This  is  what  occurs  among  animals  and  savage 
men.  Only  the  most  robust  attain  the  adult  age, 
and  these  only  reproduce  the  species.  Medicine 
and  the  aids  of  the  social  state  prolong  the  lives  of 
feeble  creatures  whose  posterity  is  usually  equally 
feeble.  Among  the  Spartans,  barbarous  regula- 
tions put  an  end  to  the  existence  of  mal-formed 
infants,  that  the  strength  and  beauty  of  the  race 
might  be  preserved.  Such  regulations  are  anti- 
pathetic to  our  customs;  nevertheless  it  might  be 
desirable  that  we  should  devote  ourselves  to  the 
preservation  of  the  human  race  from  the  causes  of 
weakness  and  degeneracy/' 

"  The  decadence  of  the  Greeks  and  Eomans 
without  change  of  race  proves  the  influence  of  in- 
stitutions upon  customs." 

We  will  give  here  a  fragment  on  political  econ- 
omy, to  show  the  variety  contained  in  the  pages  on 
which  we  draw : 

"  According  to  the  system  of  modern  economists, 
it  would  be  desirable  that  the  government  should 
interfere  as  little  as  possible  in  the  commerce  and 
industry  of  the  country.  Nevertheless  we  cannot 
deny  that  in  certain  circumstances  this  interven- 
tion is  very  useful." 


212  APPENDIX  A. 

"  Taxes  are  regarded  by  economists  as  an  evil, 
but  as  a  necessary  evil,  since  they  provide  for  pub- 
lic expenses.  Consequently,  economists  think  that 
if  the  government  possessed  sufficient  revenues,  in 
domains  for  example,  the  suppression  of  all  taxes 
would  be  a  desirable  measure." 

"  Taxes  are  a  means  of  influencing  production 
and  commerce  to  give  to  them  a  direction  which 
they  would  not  naturally  have  taken.  Such  an 
influence  may  undoubtedly  have  disagreeable  con- 
sequences if  the  taxes  are  imposed  without  dis- 
crimination or  exclusively  for  a  fiscal  purpose,  but 
it  is  entirely  otherwise  if  wisdom  and  tact  preside 
at  their  institution." 

"  A  tax  on  the  rent  of  a  farm  would  be  much 
better  than  a  tax  on  the  land  itself.  Proprietors 
then  could  only  avoid  taxes  by  themselves  improv- 
ing their  property.  As  it  is,  they  merely  collect 
the  rents,  and  usually  employ  their  surplus  in  un- 
productive expenditure,  while  the  proprietary 
farmers  voluntarily  devote  theirs  to  the  improve- 
ment of  the  land." 

"  A  tax  on  the  farms  would  then  result  in  the 
proprietors  themselves  working  the  lands,  and  tnis 
would  mean  better  cultivation,  and  improvements 
which  would  yield  returns  indeed,  but  at  too  re- 
mote a  period  for  the  tenant.  It  would  tend  to  a 


APPENDIX  A  213 

division  of  landed  property,  men  of  small  fortune 
uniting  in  the  purchase  with  capitalists  who  seek 
only  the  rent  or  payment  for  the  land." 

"  Great  capitalists  could  not  themselves  culti- 
vate vast  extents  of  land,  and  not  wanting  to  di- 
minish their  revenues  by  renting  them,  would  be 
induced  to  sell  portions  suitable  for  cultivation  by 
their  new  owners,  and  would  then  carry  their 
money  into  new  industrial  and  commercial  enter- 
prises. " 

"  The  competition  of  the  sellers  would  cause  a 
momentary  fall  in  the  price  of  the  lands,  and  would 
enable  small  farmers  to  become  land-owners.  The 
number  of  vast  estates  often  badly  managed  would 
then  be  diminished,  and  considerable  fortunes, 
changing  hands  more  easily,  would  naturally  pass 
into  those  which  would  be  most  likely  to  increase 
their  value." 

"Proprietors,  becoming  cultivators  to  escape  the 
taxes,  would  settle  in  the  country,  where  their  pres- 
ence would  disseminate  intelligence  and  comfort; 
their  revenues,  before  spent  unprofitably,  would 
then  pay  expenses  and  improvements  on  their 
propert}r." 

<(  The  establishment  of  such  a  tax  would  cer- 
tainly find  many  opponents  among  proprietors, 
landed  non-cultivators  who  form  in  fact  the  influ- 


214  APPENDIX  A. 

ential  personnel  in  the  state,  for  it  is  they  who 
usually  make  the  laws." 

"  Perhaps  it  would  be  necessary  to  weaken  their 
opposition  by  not  subjecting  the  actual  proprietors 
to  the  new  tax,  which  might  take  effect  only  with 
the  next  change  either  by  sale  or  by  inheritance. 
A  restriction  of  the  right  of  transfer  would  also 
facilitate  the  passage  from  one  situation  to  the 
other.  All  changes  in  taxes  should,  as  a  general 
thing,  be  made  gradually,  in  order  to  avoid  sudden 
changes  of  fortune." 

"  We  may  consider  the  renting  of  a  property 
for  several  years  as  a  sale  of  the  usufruct  during 
the  time  of  the  lease.  Now  nine  years'  possession, 
for  example,  is  equal  to  more  than  a  third  of  the 
value  of  the  property,  supposing  the  annual  prod- 
uct to  be  one  twentieth  of  the  capital.  It  would 
then  be  reasonable  to  apply  to  this  sort  of  sale  the 
laws  which  govern  that  of  landed  property,  and 
consequently  the  mutation  tax.  The  person  who 
cannot  or  will  not  cultivate  his  soil,  instead  of 
alienating  the  property  itself,  binds  himself  to 
alienate  the  usufruct  for  a  time,  and  the  price  is 
paid  at  stated  intervals  instead  of  all  at  once. 
There  is  farm  rent." 

"Now  it  is  by  a  fiction  that  the  purchaser  pays 
the  mutation  tax.  In  fact,  it  is  always  the  seller 


APPENDIX  A.  215 

who  pays  it.  The  buyer  compares  the  money  that 
he  spends  with  the  advantage  that  he  gains,  and 
this  comparison  determines  it.  If  he  did  not  make 
money  out  of  it  he  would  not  buy  it.  When  the 
registration  tax  did  not  exist,  the  purchaser  had  to 
pay  the  same  sum  for  the  same  purpose,  and  this 
sum  went  into  the  pocket  of  the  seller/' 

"  Proprietors  of  lands,  then,  after  all,  have  to 
bear  the  mutation  taxes.  All  increase  of  these 
taxes  is  a  loss  for  them,  and  these  taxes  are  heav- 
ier on  the  small  proprietors  than  on  the  large,  be- 
cause their  changes  are  more  frequent.  The  tax 
on  the  farms,  on  the  contrary,  would  bear  more 
heavily  on  large  estates. " 

"  The  tax  on  farms  not  affecting  the  owners  of 
timber,  would  be  made  up  by  a  tax  on  the  felling, 
a  very  justifiable  tax,  for  standing  timber  is  landed 
property.  Standing  timber  is  often  worth  much 
more  than  the  land  on  which  it  stands." 

Finally,  we  will  give  some  thoughts  which  reveal 
the  religious  sentiments  of  Sadi  Carnot: 

''Men  attribute  to  chance  those  events  of  the 
causes  of  which  they  are  ignorant.  If  they  suc- 
ceed in  divining  these  causes,  chance  disappears. 
To  say  that  a  thing  has  happened  by  chance, 


216  APPENDIX  A. 

is  to  say  that  we  have  not  been  able  to  foresee  it. 
I  do  not  myself  believe  that  any  other  acceptation 
can  be  given  to  this  word.  What  to  an  ignorant 
man  is  chance,  cannot  be  chance  to  one  better  in- 
structed." 

"If  human  reason  is  incapable  of  discovering 
the  mysteries  of  Divinity,  why  has  not  Divinity 
made  human  reason  more  clear-sighted  ?" 

"God  cannot  punish  man  for  not  believing 
when  he  could  so  easily  have  enlightened  and  con 
vinced  him." 

"If  God  is  absolutely  good,  why  should 
He  punish  the  sinner  for  all  eternity,  since 
He  does  not  lead  him  to  good,  or  give  him  an 
example  ?" 

"According  to  the  doctrine  of  the  church,  God 
resembles  a  sphinx  proposing  enigmas,  and  devour- 
ing those  who  cannot  guess  them." 

"  The  church  attributes  to  God  all  human  pas- 
sions— anger,  desire  for  vengeance,  curiosity,  tyr- 
anny, partiality,  idleness." 

"  If  Christianity  were  pruned  of  all  which  is 
not  Christ,  this  religion  would  be  the  simplest  in 
the  world." 

""What  motives  have  influenced  the  writers  who 
have  rejected  all  religious  systems  ?  Is  it  the  con- 
viction that  the  ideas  which  they  oppose  are  all 


APPENDIX  A.  217 

injurious  to  society?  Have  they  not  rather  in- 
cluded in  the  same  proscription  religion  and  the 
abuse  of  it  ?" 

"  The  belief  in  an  all-powerful  Being,  who  loves 
us  and  watches  over  us,  gives  to  the  mind  great 
strength  to  endure  misfortune." 

"  A  religion  suited  to  the  soul  and  preached  by 
men  worthy  of  respect  would  exercise  the  most 
salutary  influence  upon  society  and  customs." 


II.  NOTES  OF  SADI  CARNOT  OK  MATHEMATICS, 
PHYSICS,  AND  OTHEE  SUBJECTS. 

Up  to  the  present  time  the  changes  caused  in 
the  temperature  of  bodies  by  motion  have  been 
very  little  studied.  This  class  of  phenomena  mer- 
its, however,  the  attention  of  observers.  When 
bodies  are  in  motion,  especially  when  that  motion 
disappears,  or  when  it  produces  motive  power,  re- 
markable changes  take  place  in  the  distribution  of 
heat,  and  perhaps  in  its  quantity. 

We  will  collect  a  few  facts  which  exhibit  this 
phenomenon  most  clearly. 

1.  The  Collision  of  Bodies. — We  know  that  in 
the  collision  of  bodies  there  is  always  expenditure 
of  motive  power.  Perfectly  elastic  bodies  only  form 
an  exception,  and  none  such  are  found  in  nature. 


218  APPENDIX  A. 

We  also  know  that  always  in  the  collision  of 
bodies  there  occurs  a  change  of  temperature,  an 
elevation  of  temperature.  We  cannot,  as  did  M. 
Berthollet,  attribute  the  heat  set  free  in  this  case 
to  the  reduction  of  the  volume  of  the  body;  for 
when  this  reduction  has  reached  its  limit  the  liber- 
ation of  heat  would  cease.  Now  this  does  not  oc- 
cur. 

It  is  sufficient  that  the  body  change  form  by  per- 
cussion, without  change  of  volume,  to  produce  dis- 
engagement of  heat. 

If,  for  example,  we  take  a  cube  of  lead  and  strike 
it  successively  on  each  of  its  faces,  there  will  always 
be  heat  liberated,  without  sensible  diminution  in 
this  disengagement,  so  long  as  the  blows  are  con- 
tinued with  equal  force.  This  does  not  occur  when 
medals  are  struck.  In  this  case  the  metal  cannot 
change  form  after  the  first  blows  of  the  die,  and 
the  effect  of  the  collision  is  not  conveyed  to  the 
medal,  but  to  the  threads  of  the  screw  which  are 
strained,  and  to  its  supports. 

It  would  seem,  then,  that  heat  set  free  should 
be  attributed  to  the  friction  of  the  molecules  of 
the  metal,  which  change  place  relatively  to  each 
other,  that  is,  the  heat  is  set  free  just  where  the 
moving  force  is  expended. 

A  similar  remark  will  apply  in  regard  to  the  col- 


APPENDIX  A.  219 

lision  of  two  bodies  of  differing  hardness — lead  and 
iron  for  instance.  The  first  of  these  metals  be- 
comes very  hot,  while  the  second  does  not  vary  sen- 
sibly in  temperature.  But  the  motive  power  is 
almost  wholly  exhausted  in  changing  the  form  of 
the  first  of  these  metals.  We  may  also  cite,  as  a 
fact  of  the  same  nature,  the  heat  produced  by  the 
extension  of  a  metallic  rod  just  before  it  breaks. 
Experiment  has  proved  that,  other  things  being 
equal,  the  greater  the  elongation  before  rupture, 
the  more  considerable  is  the  elevation  of  tempera- 
ture. 

(2)  [The  remainder  is  blank.] 

When  a  hypothesis  no  longer  suffices  to  explain 
phenomena,  it  should  be  abandoned. 

This  is  the  case  with  the  hypothesis  which  re- 
gards caloric  as  matter,  as  a  subtile  fluid. 

The  experimental  facts  tending  to  destroy  this 
theory  are  as  follows : 

(1)  The  development  of  heat  by  percussion  or 
the  friction  of  bodies  (experiments  of  Rumford, 
friction  of  wheels  on  their  spindles,  on  the  axles, 
experiments  to  be  made).  Here  the  elevation  of 
temperature  takes  place  at  the  same  time  in  the 
body  rubbing  and  the  body  rubbed.  Moreover, 
they  do  not  change  perceptibly  in  form  or  nature 


220  APPENDIX  A. 

(to  be  proved).  Thus  heat  is  produced  by  motion. 
If  it  is  matter,  it  must  be  admitted  that  the  matter 
is  created  by  motion. 

(2)  When  an  air-pump  is  worked,  and  at  the 
same  time  air  is  admitted  into  the  receiver,  the 
temperature  remains  constant  in  the  receiver.    It 
remains  constant  on  the   outside.     Consequently, 
the  air  compressed  by  the  pumps  must  rise   in 
temperature    above  the  air  outside,  and  it  is  ex- 
pelled at  a  higher  temperature.     The  air   enters 
then  at  a  temperature  of   10°,  for  instance,  and 
leaves  at  another,  10°  -f  90°  or  100°,  for  example. 
Thus  heat  has  been  created  by  motion. 

(3)  If  the  air  in  a  reservoir  is  compressed,  and  at 
the  same  time  allowed  to  escape  through  a  little 
opening,  there  is  by  the  compression  elevation  of 
temperature,  by  the  escape  lowering  of  tempera- 
ture  (according  to  Gay-Lussac  and  Welter).     The 
air  then  enters  at  one  side  at  one  temperature  and 
escapes  at  the  other  side  at  a  higher  temperature, 
from  which  follows  the  same  conclusion  as  in  the 
preceding  case. 

(Experiment  to  be  made :  To  fit  to  a  high-pres- 
sure boiler  a  cock  and  a  tube  leading  to  it  and  empty- 
ing into  the  atmosphere;  to  open  the  cock  a  little 
way,  and  present  a  thermometer  to  the  outlet  of 
the  steam;  to  see  if  it  remains  at  100°  or  more; 


APPENDIX  A. 


to  see  if  steam  is  liquefied   in  the  pipe;  to  see 
whether  it  comes  out  cloudy  or  transparent.) 

(4)  The  elevation    of  temperature  which  takes 
place  at  the  time  of  the  entrance  of  the  air  into  the 
vacuum,  an  elevation  that  cannot  be  attributed  to 
the  compression  of  the  air  remaining  (air  which 
may  be  replaced  by  steam),  can  therefore  be  at- 
tributed only  to  the  friction  of  the  air  against  the 
walls  of  the  opening,  or  against  the  interior  of  the 
receiver,  or  against  itself. 

(5)  M.   Gay-Lussac  showed  (it  is  said)  that  if 
two   receivers  were   put    in   communication  with 
each  other,  the  one  a  vacuum,  the  other  full  of  air, 
the  temperature  would  rise  in  one  as   much  as  it 
would  fall  in   the  other.     If,  then,  both  be  com- 
pressed one  half,  the  first  would  return  to  its  pre- 
vious temperature  and  the  second  to  a  much  higher 
one.     Mixing  them,  the   whole   mass    would   be 
heated. 

When  the  air  enters  a  vacuum,  its  passage 
through  one  small  opening  and  the  motion  im- 
parted to  it  in  the  interior  appear  to  produce  ele- 
vation of  temperature. 

We  may  be  allowed  to  express  here  an  hypothe- 
sis in  regard  to  the  nature  of  heat. 

At  present,  light  is  generally  regarded  as  the 


222  APPENDIX  A. 

result  of  a  vibratory  movement  of  the  ethereal 
fluid.  Light  produces  heat,  or  at  least  accompa- 
nies the  radiating  heat,  and  moves  with  the  same 
velocity  as  heat.  Eadiating  heat  is  then  a  vibratory 
movement.  It  would  be  ridiculous  to  suppose  that 
it  is  an  emission  of  matter  while  the  light  which 
accompanies  it  could  be  only  a  movement. 

Could  a  motion  (that  of  radiating  heat)  pro- 
duce matter  (caloric)  ? 

No,  undoubtedly;  it  can  only  produce  a  motion. 
Heat  is  then  the  result  of  a  motion. 

Then  it  is  plain  that  it  could  be  produced  by  the 
consumption  of  motive  power,  and  that  it  could 
produce  this  power. 

All  the  other  phenomena — composition  and  de- 
composition of  bodies,  passage  to  the  gaseous  state, 
specific  heat,  equilibrium  of  heat,  its  more  or  less 
easy  transmission,  its  constancy  in  experiments 
with  the  calorimeter — could  be  explained  by  this 
hypothesis.  But  it  would  be  difficult  to  explain 
why,  in  the  development  of  motive  power  by  heat, 
a  cold  body  is  necessary ;  why,  in  consuming  the 
heat  of  a  warm  body,  motion  cannot  be  produced. 

It  appears  very  difficult  to  penetrate  into  the 
real  essence  of  bodies.  To  avoid  erroneous  reason- 
ing, it  would  be  necessary  to  investigate  carefully 


APPENDIX  A.  223 

the  source  of  our  knowledge  in  regard  to  the  na- 
ture of  bodies,  their  form,  their  forces;  to  see  what 
the  primitive  notions  are,  to  see  from  what  im- 
pressions they  are  derived ;  to  see  how  one  is  raised 
successively  to  the  different  degrees  of  abstraction. 

Is  heat  the  result  of  a  vibratory  motion  of  mole- 
cules ?  If  this  is  so,  quantity  of  heat  is  simply 
quantity  of  motive  power.  As  long  as  motive 
power  is  employed  to  produce  vibratory  movements, 
the  quantity  of  heat  must  be  unchangeable;  which 
seems  to  follow  from  experiments  with  the  calo- 
rimeter; but  when  it  passes  into  movements  of  sen- 
sible extent,  the  quantity  of  heat  can  no  longer 
remain  constant. 

Can  examples  be  found  of  the  production  of 
motive  power  with  actual  consumption  of  heat  ? 
It  seems  that  we  may  find  production  of  heat  with 
consumption  of  motive  power  (re-entrance  of  the 
air  into  a  vacuum,  for  example). 

What  is  the  cause  of  the  production  of  heat  in 
combinations  of  substances?  What  is  radiant 
caloric  ? 

Liquefaction  of  bodies,  solidification  of  liquids, 


'224.  APPENDIX  A. 

crystallization — are  they  not  forms  of  combinations 
of  integrant  molecules  ? 

Supposing  heat  due  to  a  vibratory  movement, 
how  can  the  passage  from  the  solid  or  the  liquid  to 
the  gaseous  state  be  explained  ? 

When  motive  power  is  produced  by  the  passage 
of  heat  from  the  body  A  to  the  body  B,  is  the  quan- 
tity of  this  heat  which  arrives  at  B  (if  it  is  not  the 
same  as  that  which  has  been  taken  from  A,  if  a 
portion  has  really  been  consumed  to  produce  mo- 
tive power)  the  same  whatever  may  be  the  sub- 
stance employed  to  realize  the  motive  power? 

Is  there  any  way  of  using  more  heat  in  the  pro- 
duction of  motive  power,  and  of  causing  less  to 
reach  the  body  B  ?  Could  we  even  utilize  it  en- 
tirely, allowing  none  to  go  to  the  body  B  ?  If 
this  were  possible,  motive  power  could  be  created 
without  consumption  of  combustible,  and  by  mere 
destruction  of  the  heat  of  bodies. 

Is  it  absolutely  certain  that  steam  after  having 
operated  an  engine  and  produced  motive  power 
can  raise  the  temperature  of  the  water  of  conden- 
sation as  if  it  had  been  conducted  directly  into  it? 

Reasoning  shows  us  that  there  cannot  be  loss  of 


APPENDIX  A.  225 

living  force,  or,  which  is  the  same  thing,  of  motive 
power,  if  the  bodies  act  upon  each  other  without 
directly  touching  each  other,  without  actual  col- 
lision. Now  everything  leads  us  to  believe  that 
the  molecules  of  bodies  are  always  separated  from 
each  other  by  some  space,  that  they  are  never  ac- 
tually in  contact.  If  they  touched  each  other, 
they  would  remain  united,  and  consequently 
change  form. 

If  the  molecules  of  bodies  are  never  in  close  con- 
tact with  each  other  whatever  may  be  the  forces 
which  separate  or  attract  them,  there  can  never 
be  either  production  or  loss  of  motive  power  in 
nature.  This  power  must  be  as  unchangeable  in 
quantity  as  matter.  Then  the  direct  re-establish- 
ment of  equilibrium  of  the  caloric,  and  its  re-estab- 
lishment with  production  of  motive  power,  would 
be  essentially  different  from  each  other. 

Heat  is  simply  motive  power,  or  rather  motion 
which  has  changed  form.  It  is  a  movement  among 
the  particles  of  bodies.  Wherever  there  is  destruc- 
tion of  motive  power  there  is,  at  the  same  time, 
production  of  heat  in  quantity  exactly  proportional 
to  the  quantity  of  motive  power  destroyed.  Ke- 
ciprocally,  wherever  there  is  destruction  of  heat, 
there  is  production  of  motive  power. 


226  APPENDIX  A. 

We  can  then  establish  the  general  proposition 
that  motive  power  is,  in  quantity,  invariable  in 
nature;  that  it  is,  correctly  speaking,  never  either 
produced  or  destroyed.  It  is  true  that  it  changes 
form,  that  is,  it  produces  sometimes  one  sort  of 
motion,  sometimes  another,  but  it  is  never  annihi- 
lated. 

According  to  some  ideas  that  I  have  formed 
on  the  theory  of  heat,  the  production  of  a  unit  of 
motive  power  necessitates  the  destruction  of  2.70 
units  of  heats. 

A  machine  which  would  produce  20  units  of 
motive  power  per  kilogram  of  coal  ought  to  destroy 
20  X  2.70 


7000 


of  the  heat  developed  by  the  combustion. 


20  X  2.70         8       ,  ,.    ,  .     .  1 

about>  that  1S> less  than 


7000  1000 

(Each  unit  of  motive  power,  or  dyname,  repre- 
senting the  weight  of  one  cubic  metre  of  water 
raised  to  the  height  of  one  metre.) 

Experiments  to  be  made  on  Heat  and  Motive  Power. 

To  repeat  Rumford's  experiments  in  the  drilling 
of  a  metal  in  water,  but  to  measure  the  motive 
power  consumed  at  the  same  time  as  the  heat  pro- 


APPENDIX  A.  227 

duced;   same  experiments  on  several  metals  and 
on  wood. 

To  strike  a  piece  of  lead  in  various  ways,  to 
measure  the  motive  power  consumed  and  the  heat 
produced.  Same  experiments  on  other  metals. 

To  strongly  agitate  water  in  a  small  cask  or  in 
a  double-acting  pump  having  a  piston  pierced  with 
a  small  opening. 

Experiment  of  the  same  sort  on  the  agitation  of 
mercury,  alcohol,  air  and  other  gases.  To  measure 
the  motive  power  consumed  and  heat  produced. 

To  admit  air  into  a  vacuum  or  into  air  more  or 
less  rarefied;  id.  for  other  gases  or  vapors.  To 
examine  the  elevation  of  temperature  by  means  of 
the  manometer  and  the  thermometer  of  Breguet. 
Estimation  of  the  error  of  the  thermometer  in  the 
time  required  for  the  air  to  vary  a  certain  number 
of  degrees.  These  experiments  would  serve  to 
measure  the  changes  which  take  place  in  the  tem- 
perature of  the  gas  during  its  changes  of  volume. 
They  would  also  furnish  means  of  comparing  these 
changes  with  the  quantities  of  motive  power  pro- 
duced or  consumed. 

Expel  the  air  from  a  large  reservoir  in  which  it  is 
compressed,  and  check  its  velocity  in  a  large  pipe  in 


228  APPENDIX  A. 

which  solid  bodies  have  been  placed;  measure  the 
temperature  when  it  has  become  uniform.  See  if 
it  is  the  same  as  in  the  reservoir.  Same  experi- 
ments with  other  gases  and  with  vapor  formed 
under  different  pressures. 

To  repeat  Dalton's  experiments  and  carry  them 
on  to  pressures  of  thirty  or  forty  atmospheres.  To 
measure  the  constituent  heat  of  the  vapor  within 
these  limits. 

Id.  on  the  vapor  of  alcohol,  of  ether,  of  essence 
of  turpentine,  of  mercury,  to  prove  whether  the 
agent  employed  makes  any  difference  in  the  pro- 
duction of  motive  power. 

Id.  on  water  charged  with  a  deliquescent  salt, 
the  calcium  chloride,  for  instance. 

Is  the  law  of  tensions  always  the  same?  To 
measure  the  specific  heat  of  vapor. 

Experiments  to  le  made  on  the  Tension  of  Vapors. 

A  graduated  capillary  tube  filled  with  water, 
mercury,  or  with  oil  and  air.  Plunge  this  tube 
into  a  bath  of  oil,  of  mercury,  or  of  melted  lead. 
To  measure  the  temperature  by  an  air  thermometer. 

Same  experiments  with  alcohol,  ether,  sulphide 
of  carbon,  muriatic  ether,  essence  of  turpentine, 
sulphur,  phosphorus. 


APPENDIX  A.  229 

Experiments  on  the  tension  of  steam  with  a 
boiler,  and  a  thermometric  tube  full  of  air.  A 
thermometer  will  be  placed  in  a  tube  immersed  in 
the  boiler,  open  outwards  and  filled  with  oil  or 
mercury. 

Experiments  by  means  of  a  simple  capillary 
tube  filled  with  three  successive  parts — first  of  air, 
second  of  mercury,  third  of  water  or  other  liquid 
of  which  the  tension  can  be  measured  (of  alcohol, 
of  ether,  of  essence  of  turpentine,  of  lavender,  of 
sulphide  of  carbon,  of  muriatic  ether,  etc.). 
One  end  of  the  tube  may  be  immersed  in  a  bath 
of  mercury  or  oil,  the  temperature  of  which  is  to 
be  measured.  The  column  of  mercury  can  be  made 
long  enough  to  allow  of  the  air  being  previously 
compressed  or  rarefied. 


FIG.  6. 

The  tube  will  be  bent  into  a  spiral  at  one  end, 
the  straight  part  being  graduated  (thus  permitting 
the  tension  of  mercurial  vapor  to  be  measured). 

Experiments  on  the  tension  of  vapors  at  low 


230  APPENDIX  A. 

temperature,  with  a  thermometric  tube  bent 
round,  and  filled  partly  with  mercury, 
partly  with  water  or  alcohol.  The  mer- 
cury will  operate  by  its  weight.  The 
upper  part  of  the  tube  will  be  empty  and 
sealed,  or  fully  open  to  the  atmosphere. 

The  bulb  will  be  immersed  in  water  the 
temperature  of  which  is  to  be  measured. 
.  7.     If  the   tube   is    sealed,  the  upper  part 
must  be  cooled. 

The  bulb  might  contain  water,  ether,  or  essence 
of  turpentine. 

If  the  tube  is  sealed,  the  tension  of  mercurial 
vapor  could  be  measured. 

Experiments  on  the  constituent  heat  of  vapors 
by  means  of  a  barometric  tube  having  two  en- 
larged bulbs.  One  of  the  bulbs  may  be  im- 
mersed in  cold  water,  and  the  elevation  of  temper- 
ature of  this  water  will  indicate  the  constituent 
heat  of  the  vapor. 


FIG.  8. 


APPENDIX  A.  231 

The  other  bulb  may  be  warmed  either  by  boiling 
liquid  or  by  fire. 

Water,  alcohol,  steam,  ether,  mercury,  acetic 
acid,  sulphide  of  carbon. 

The  operation  may  be  repeated  and  add  the  results. 

Experiments  to  le  made  on  Oases  and  Vapors. 

To  measure  the  temperature  acquired  by  the  air 
introduced  into  a  vacuum  or  space  containing  pre- 
viously rarefied  air. 

If  the  vacuum  is  made  under  the  glass  receiver 
of  an  air-pump,  and  the  cock  admitting  the  outer 
air  be  tsuddenly  opened,  the  introduction  of  this 
air  will  cause  a  Breguet  thermometer  to  rise  to  50° 
or  60°.  To  examine  the  movement  of  this 
thermometer  when  the  reintroduction 
takes  place  only  by  degrees,  to  compare 
it  with  the  movement  of  the  manometer. 

Construction  of  a  manometer  which  15] 
may  give  the  pressure  almost  instanta- 
neously. 

Imagine  a  capillary  tube  bent  into  a 
spiral  at  one  end,  and  having  one  ex- 
tremity closed,  the  other  open.  This 
tube  will  be  perfectly  dry  and  a  small 
index  of  mercury  may  be  introduced 
into  it.  The  diameter  of  the  tube  will  be  small 


232  APPENDIX  A. 

enough  for  the  air  enclosed  in  it  to  take  almost 
instantly  the  temperature  of  the  glass.  We  shall 
try  to  ascertain  the  time  necessary  for  the  estab- 
lishment of  this  equilibrium  of  temperature  by 
placing  the  tube  under  the  receiver  of  the  air- 
pump,  making  a  partial  vacuum,  and  admitting 
the  air.  We  shall  see  whether,  some  seconds  after 
the  introduction,  the  index  perceptibly  moves. 
The  index  must  be  of  very  light  weight  to  avoid 
oscillation  as  much  as  possible. 

For  the  same  reason,  the  capillary  tube  should 
be  also  as  narrow  as  possible.  If  the  straight  part 
of  the  tube  is  equal  to  the  bent  part  and  the  index 
be  placed  at  the  beginning  of  the  bent  part,  for  a 
pressure  equal  to  atmospheric  pressure,  it  would 
not  be  necessary  to  subject  the  instrument  to  a 
less  pressure  than  -J  atmosphere.  It  is  between 
these  two  limits  that  it  would  serve  as  a  measure. 

It  might  end  in  an  open  enlargement  to  prevent 
the  projection  of  the  mercury  outside  the  tube. 
Disposed  in  this  way,  it  could  be  used  as  a  general 
measure  for  pressures  between  p  and  £_/?;  p  being 
anything  whatever.  The  apparatus  will  be  fast- 
ened to  a  board  bearing  a  graduated  scale  placed 
against  the  straight  tube.  The  scale  will  be,  for 
instance,  numbered  by  fives  or  tens.  A  correspond- 
ing table  denoting  pressures  would  be  required. 


APPENDIX  A.  233 

Placing  the  instrument  under  the  receiver  and 
forming  a  partial  vacuum,  the  index  will  rise  into 
the  enlargement.  Then,  admitting  the  air  by  de- 
grees and  very  slowly,  we  may  note  the  correspond- 
ence between  the  heights  of  the  ordinary  mercury 
manometer  and  the  point  which  will  be  reached 
by  the  lower  face  of  the  index  of  the  instrument. 
This  will  answer  to  form  a  comparative  table  of 
the  pressures  and  the  numbers  of  the  scale.  The 
pressures  would  be  represented  by  their  relations 
to  the  observed  pressure  at  the  moment  of  the 
passage  of  the  index  over  zero,  for  any  other  fixed 
number  of  the  scale. 

Thus,  for  example,  suppose  that  we  observed  on 
the  manometer  400  or  n  millimetres  of  mercury 
when  the  index  is  on  o,  then  n'  when  the  index  is 
on  1,  n"  when  on  2,  and  so  on.  This  will  give  the 

n'   n" 

ratios  —,—,...  which  must  be  inscribed  in  the 
n    n 

table.  Then  n  could  be  varied  at  pleasure,  and 
the  table  could  still  be  used. 

In  fact,  according  to  the  law  of  Mariotte,  vol- 
umes preserving  the  same  ratios,  pressures  should 
also  preserve  the  same  ratios  to  each  other. 

Let  p  be  the  pressure  when  the  index  is  on  o,  v 
the  volume  of  air  at  the  same  moment,  p'  and  vf 
the  same  pressures  and  volume  at  the  moment 


234  APPENDIX  A. 

when  the  index  is  on  1.     Whether  the  air  be  ex- 
pelled or  admitted  the  pressures  would  be  instead 
of  p  and^/,  q  and  qr.     But  there  would  follow 
p  :  p'  :  :  v'  :  v        and          q  :  q'  :  :  v'  :  v ; 
then  p  :p'  :  :  q  :  q'. 

We  should  moreover  work  at  a  uniform  tempera- 
ture and  note  the  variations. 

If  the  straight  part  of  the  tube  were  perfectly 
calibrated,  the  volumes,  and  consequently  the  pres- 
sures, would  form  a  geometrical  progression,  when 
the  figures  of  the  scale  would  be  found  to  be  in 
arithmetical  progression,  and  a  table  of  logarithms 
would  enable  one  to  be  found  from  the  other. 

In  order  to  increase  as  required  the  mass  of  air 
enclosed  in  the  tube  the  instrument  must  be 
placed  on  its  side  or  flat,  in  the  air-pump  receivers. 
The  mercury  index  would  be  placed  in  the  lateral 
part  of  the  enlargement  of  the  tube,  and  the  at- 
mospheric air  would  enter.  The  instrument 
might  also  be  heated  in  this  position. 

Care  must  be  taken  to  admit  only  very  dry  air, 
which  could  be  obtained  by  placing  under  the  re- 
ceiver calcium  chloride  or  any  other  substance 
which  absorbs  moisture  greedily. 

Instead  of  bending  the  tube  into  a  spiral,  it 
might  be  bent  in  the  middle  in  the  form  of  a  U, 
or  it  might  be  better  to  form  three,  four  or  mors 


APPENDIX  A.  235 

parallel  branches.  Making  the  tube  very  long,  the 
index  would  have  a  larger  range  for  the  same 
changes  of  pressure,  and  the  results  produced 
could  then  be  measured  by  a  slight  variation  in 
density  in  the  air  of  the  receiver. 

Comparison  of  the  Rapidity  with  which  the  Air 
cools  in  the  Receiver  and  in  the  Tube. 

Let  us  suppose,  what  I  believe  to  be  very  near 
the  truth,  that  the  heat  absorbed  is  proportional 
to  the  surface  of  the  bodies  in  contact.  From 
this  we  can  infer  without  difficulty,  that  the  rapid- 
ity of  the  cooling  of  the  air  in  two  cylindrical 
tubes  would  be  inversely  as  their  diameters. 

If  the  receiver  is  considered  as  a  tube  of  two 
decimetres  in  diameter,  and  the  manometer  as  a 
tube  of  one  millimetre  diameter,  the  rapidity  of 
the  cooling  of  the  air  would  be  in  the  ratio,  very 
nearly,  of  1  to  200. 

Extent  of  the  Movement  of  the  Index. 

Suppose  the  tube  turned  up  on  itself  five  times 
and  having  a  total  length  of  1  metre;  a  variation 
of  density  equal  to  TV  in  the  air  will  give  a  move- 
ment of  1  decimetre;  a  variation  of  heat  of  1  de- 
gree supposed  to  be  equivalent  to  a  variation  of 
density  of  ^  will  give  ^  of  a  metre,  or  about 


236  APPENDIX  A. 

3mm.70,  quite  an  appreciable  quantity.  As  to  the 
time  required  to  move  the  mercury  index,  regard 
being  had  to  its  mass,  if  we  suppose  it  1  centi- 
metre long,  and  the  variation  of  pressure  TJ^  of  an 
atmosphere,  it  would  require  about  £  of  a  second 
to  make  it  pass  over  one  decimetre. 

Use  of  the  Instrument  in  Measuring  the  Varia- 
tions of  the  Tensions  of  the  Air  under  a  Pneu- 
matic Receiver. 

At  each  stroke  of  the  piston  which  expands  the 
air  under  the  pneumatic  receiver  when  a  vacuum 
is  to  be  created,  a  lowering  of  pressure  is  produced, 
and  undoubtedly  a  change  of  temperature.  It  can 
be  determined  approximately,  at  least,  by  observing 
the  position  of  the  manometer  at  the  instant  after 
the  dilatation  has  taken  place,  and  again  after  a 
time  long  enough  for  the  temperature  to  have  re- 
turned to  its  original  point,  that  of  the  surrounding 
bodies.  Comparison  of  the  elastic  force  in  the  two 
cases  will  lead  to  comparison  of  the  temperatures. 

The  temperature  having  returned  to  its  original 
point,  we  will  give  a  second  stroke  of  the  piston 
which  will  rarefy  the  air  more  than  the  former, 
and  thus  we  will  make  two  observations  of  the 
manometer,  before  and  after  the  return  to  the 
former  temperature.  And  so  on. 


OF  THE 

UNIVERSITY 


APPENDIX  B. 

CARNOT'S    FOOT-NOTES. 

NOTE  A. — The  objection  may  perhaps  be  raised 
here,  that  perpetual  motion,  demonstrated  to  be 
impossible  by  mechanical  action  alone,  may  pos- 
sibly not  be  so  if  the  power  either  of  heat  or  elec- 
tricity be  exerted;  but  is  it  possible  to  conceive 
the  phenomena  of  heat  and  electricity  as  due  to 
anything  else  than  some  kind  of  motion  of  the 
body,  and  as  such  should  they  not  be  subjected  to 
the  general  laws  of  mechanics  ?  Do  we  not  know 
besides,  a  posteriori,  that  all  the  attempts  made  to 
produce  perpetual  motion  by  any  means  whatever 
have  been  fruitless  ? — that  we  have  never  succeeded 
in  producing  a  motion  veritably  perpetual,  that 
is,  a  motion  which  will  continue  forever  without 
alteration  in  the  bodies  set  to  work  to  accomplish 
it  ?  The  electromotor  apparatus  (the  pile  of  Volta) 
has  sometimes  been  regarded  as  capable  of  pro- 
ducing perpetual  motion ;  attempts  'have  been 
made  to  realize  this  idea  by  constructing  dry  piles 
said  to  be  unchangeable  ;  but  however  it  has  been 
done,  the  apparatus  has  always  exhibited  sensible 

337 


238  APPENDIX  B. 

deteriorations  when  its  action  has  been  sustained 
for  a  time  with  any  energy. 

The  general  and  philosophic  acceptation  of  the 
words  perpetual  motion  should  include  not  only  a 
motion  susceptible  of  indefinitely  continuing  itself 
after  a  first  impulse  received,  but  the  action  of  an 
apparatus,  of  any  construction  whatever,  capable 
of  creating  motive  power  in  unlimited  quantity, 
capable  of  starting  from  rest  all  the  bodies  of  na- 
ture if  they  should  be  found  in  that  condition,  of 
overcoming  their  inertia;  capable,  finally,  of  find- 
ing in  itself  the  forces  necessary  to  move  the  whole 
universe,  to  prolong,  to  accelerate  incessantly,  its 
motion.  Such  would  be  a  veritable  creation  of 
motive  power.  If  this  were  a  possibility,  it  would 
be  useless  to  seek  in  currents  of  air  and  water  or 
in  combustibles  this  motive  power.  We  should 
have  at  our  disposal  an  inexhaustible  source  upon 
-which  we  could  draw  at  will. 

NOTE  B. — The  experimental  facts  which  best 
prove  the  change  of  temperature  of  gases  by  com- 
pression or  dilatation  are  the  following: 

(1)  The  fall  of  the  thermometer  placed  under 
the  receiver  of  a  pneumatic  machine  in  which  a 
vacuum  has  been  produced.  This  fall  is  very  sen- 
sible on  the  Breguet  thermometer:  it  may  exceed 
40°  or  50°.  The  mist  which  forms  in  this  case 


APPENDIX  B.  239 

seems  to  be  due  to  the  condensation  of  the  watery 
vapor  caused  by  the  cooling  of  the  air. 

(2)  The  inflammation  of  German  tinder  in  the 
so-called  pneumatic  tinder-boxes ;   which  are,  as 
we  know,  little  pump- chambers  in  which  the  air  is 
rapidly  compressed. 

(3)  The  fall  of  a  thermometer  placed  in  a  space 
where  the  air  has  been  first  compressed  and  then 
allowed  to  escape  by  the  opening  of  a  cock. 

(4)  The  results  of  experiments  on  the  velocity 
of   sound.      M.   de   Laplace   has  shown  that,    in 
order  to  secure  results  accurately  by  theory  and 
computation,  it  is  necessary  to  assume  the  heating 
of  the  air  by  sudden  compression. 

The  only  fact  which  may  be  adduced  in  opposi- 
tion to  the  above  is  an  experiment  of  MM.  Gay- 
Lussac  and  Welter,  described  in  the  Annales  de 
Chimie  et  de  Physique.  A  small  opening  having 
been  made  in  a  large  reservoir  of  compressed  air, 
and  the  ball  of  a  thermometer  having  been  intro- 
duced into  the  current  of  air  which  passes  out 
through  this  opening,  no  sensible  fall  of  the  tem- 
perature denoted  by  the  thermometer  has  been 
observed. 

Two  explanations  of  this  fact  may  be  given: 
(1)  The  striking  of  the  air  against  the  walls  of  the 
opening  by  which  it  escapes  may  develop  heat  in. 


240  APPENDIX  B. 

observable  quantity.  (2)  The  air  which  has  jusl 
touched  the  bowl  of  the  thermometer  possibly 
takes  again  by  its  collision  with  this  bowl,  or 
rather  by  the  effect  of  the  detour  which  it  is 
forced  to  make  by  its  rencounter,  a  density  equal 
to  that  which  it  had  in  the  receiver, — much  as  the 
water  of  a  current  rises  against  a  fixed  obstacle, 
above  its  level. 

The  change  of  temperature  occasioned  in  the 
gas  by  the  change  of  volume  may  be  regarded  as 
one  of  the  most  important  facts  of  Physics,  be- 
cause of  the  numerous  consequences  which  it 
entails,  and  at  the  same  time  as  one  of  the  most 
difficult  to  illustrate,  and  to  measure  by  decisive 
experiments.  It  seems  to  present  in  some  respects 
singular  anomalies. 

Is  it  not  to  the  cooling  of  the  air  by  dilatation 
that  the  cold  of  the  higher  regions  of  the  atmos- 
phere must  be  attributed?  The  reasons  given 
heretofore  as  an  explanation  of  this  cold  are  en- 
tirely insufficient;  it  has  been  said  that  the  air  of 
the  elevated  regions  receiving  little  reflected  heat 
from  the  earth,  and  radiating  towards  celestial 
space,  would  lose  caloric,  and  that  this  is  the  cause 
of  its  cooling;  but  this  explanation  is  refuted  by 
the  fact  that,  at  an  equal  height,  cold  reigns  with 
equal  and  even  more  intensity  on  the  elevated 


APPENDIX  B.  241 

plains  than  on  the  summit  of  the  mountains,  or  in 
those  portions  of  the  atmosphere  distant  from  the 
sun. 

NOTE  C. — We  see  no  reason  for  admitting,  a 
prioriy  the  constancy  of  the  specific  heat  of  bodies 
at  different  temperatures,  that  is,  to  admit  that 
equal  quantities  of  heat  will  produce  equal  incre- 
ments of  temperature,  when  this  body  changes 
neither  its  state  nor  its  density;  when,  for  example, 
it  is  an  elastic  fluid  enclosed  in  a  fixed  space. 
Direct  experiments  on  solid  and  liquid  bodies  have 
proved  that  between  zero  and  100°,  equal  incre- 
ments in  the  quantities  of  heat  would  produce 
nearly  equal  increments  of  temperature.  But  the 
more  recent  experiments  of  MM.  Dulong  and 
Petit  (see  Annales  de  Chimie  et  de  Physique  ,^ob- 
ruary,  March,  and  April,  1818)  have  shown  that  this 
correspondence  no  longer  continues  at  tempera- 
tures much  above  100°,  whether  these  temperatures 
be  measured  on  the  mercury  thermometer  or  on 
the  air  thermometer. 

Not  only  do  the  specific  heats  not  remain  the 
same  at  different  temperatures,  but,  also,  they  do 
not  preserve  the  same  ratios  among  themselves,  so 
that  no  thermometric  scale  could  establish  the  con- 
stancy of  all  the  specific  heats.  It  would  have  been 
interesting  to  prove  whether  the  same  irregulari- 


242  APPENDIX  B. 

ties  exist  for  gaseous  substances,  but  such  experi- 
ments presented  almost  insurmountable  difficul- 
ties. 

The  irregularities  of  specific  heats  of  solid  bodies 
might  have  been  attributed,  it  would  seem,  to  the 
latent  heat  employed  to  produce  a  beginning  of 
fusion — a  softening  which  occurs  in  most  bodies 
long  before  complete  fusion.  We  might  support 
this  opinion  by  the  following  statement:  According 
to  the  experiments  of  MM.  Dulong  and  Petit,  the 
increase  of  specific  heat  with  the  temperature  is 
more  rapid  in  solids  than  in  liquids,  although  the 
latter  possess  considerably  more  dilatability.  The 
cause  of  irregularity  just  referred  to,  if  it  is  real, 
would  disappear  entirely  in  gases. 

NOTE  D. — In  order  to  determine  the  arbitrary 
constants  A,  B,  A',  B' ,  in  accordance  with  the 
results  in  M.  Dalton's  table,  we  must  begin  by  com- 
puting the  volume  of  the  vapor  as  determined  by 
its  pressure  and  temperature, — a  result  which  is 
easily  accomplished  by  reference  to  the  laws  of 
Mariotte  and  Gay-Lussac,  the  weight  of  the  vapor 
being  fixed. 

The  volume  will  be  given  by  the  equation 

267 +  tf 

v  =  c • — , 

P 
in  which  v  is  this  volume,  t  the  temperature,  p  the 


APPENDIX  S. 


243 


pressure,  and  c  a  constant  quantity  depending  on 
the  weight  of  the  vapor  -and  on  the  units  chosen. 
We  give  here  the  table  of  the  volumes  occupied  by 
a  gramme  of  vapor  formed  at  different  tempera- 
tures,, and  consequently  under  different  pressures. 


t 

P 

V 

or  degrees  Centi- 
grade. 

or  tension  of  the  vapor 
expressed  in  millime- 
tres of  mercury. 

or  volume  of  a  gramme 
of  vapor  expressed 
in  litres. 

0 

mm. 

lit. 

0 

5.060 

185.0 

20 

17.32 

58.2 

40 

53.00 

20.4 

60 

144.6 

7.96 

80 

352.1 

3.47 

100 

760.0 

1.70 

The  first  two  columns  of  this  table  are  taken 
from  the  Traite  de  Physique  of  M.  Biot  (vol.  i.,  p. 
272  and  531).  The  third  is  calculated  by  means 
of  the  above  formula,  and  in  accordance  with  the 
result  of  experiment,  indicating  that  water  vapor- 
ized under  atmospheric  pressure  occupies  a  space 
1700  times  as  great  as  in  the  liquid  state. 

By  using  three  numbers  of  the  first  column  and 
three  corresponding  numbers  of  the  third  column, 
we  can  easily  determine  the  constants  of  our  equa- 
tion 

A  +  B  log  v 

~  A'  +  B'  log  v 


244  APPENDIX  B. 

We  will  not  enter  into  the  details  of  the  calcula- 
tion necessary  to  determine  these  quantities.  It 
is  sufficient  to  say  that  the  following  values, 


A'  =  19.64, 
B=  -1000,         B'  =    3.30, 

satisfy  fairly  well  the  prescribed  conditions,  so  that 
the  equation 

_  2268  -  1000  log  v 
'  19.  64  +  3.30  log  v 

expresses  very  nearly  the  relation  which  exists  be- 
tween the  volume  of  the  vapor  and  its  tempera- 
ture. We  may  remark  here  that  the  quantity  B' 
is  positive  and  very  small,  which  tends  to  confirm 
this  proposition  —  that  the  specific  heat  of  an  elastic 
fluid  increases  with  the  volume,  but  follows  a  slow 
progression. 

NOTE  E.  —  Were  we  to  admit  the  constancy  of 
the  specific  heat  of  a  gas  when  its  volume  does  not 
change,  but  when  its  temperature  varies,  analysis 
would  show  a  relation  between  the  motive  power 
and  the  thermometric  degree.  We  will  show  how 
this  is,  and  this  will  also  give  us  occasion  to  show 
how  some  of  the  propositions  established  above 
should  be  expressed  in  algebraic  language. 

Let  r  be  the  quantity  of  motive  power  produced 
by  the  expansion  of  a  given  quantity  of  air  passing 


APPENDIX  3.  245 

from  the  volume  of  one  litre  to  the  volume  of  v 
litres  under  constant  temperature.  If  v  increases 
by  the  infinitely  small  quantity  dv,  r  will  increase 
by  the  quantity  dr,  which,  according  to  the  nature 
of  motive  power,  will  be  equal  to  the  increase  dv 
of  volume  multiplied  by  the  expansive  force  which 
the  elastic  fluid  then  possesses;  let  p  be  this  ex- 
pansive force.  We  should  have  the  equation 

dr  =  pdv (1) 

Let  us  suppose  the  constant  temperature  under 
which  the  dilatation  takes  place  equal  to  t  degrees 
Centigrade.  If  we  call  q  the  elastic  force  of  the 
air  occupying  the  volume  1  litre  at  the  same  tem- 
perature tt  we  shall  have,  according  to  the  law  of 
Mariotte, 

-  =  "    whence    p  =  -. 
Ip  v 

If  now  P  is  the  elastic  force  of  this  same  air  at  the 
constant  volume  1,  but  at  the  temperature  zero, 
we  shall  have,  according  to  the  rule  of  M.  Gay- 
Lussac, 


whence 

P    267 


246  APPENDIX  S. 


p 

If,  to  abridge,  we  call  N  the  quantity  1^E>  the 

~ 


equation  would  become 

^  t  +  267 

P  =  N-±^—> 

whence  we  deduce,  according  to  equation  (I), 


ar, 

dr  =  N  —  •  --  dv. 

v 

Regarding  t  as  constant,  and  taking  the  integral  of 
the  two  numbers,  we  shall  have 

r  =  N(t  +  267)  log  v  +  C. 

If  we  suppose  r  =  0  when  v  =  1,  we  shall  have 
(7=0;  whence 

r  =  N(t  +  267)  log  v.     .     .     .     (2) 

This  is  the  motive  power  produced  by  the  expan- 
sion oi  the  air  which,  under  the  temperature  t,  has 
passed  from  the  volume  1  to  the  volume  v.  If  in- 
stead of  working  at  the  temperature  t  we  work  in 
precisely  vtto  name  manner  at  the  temperature 
t  -j-  dt,  the  power  developed  will  be 

r  +  dr  =  N(t  +  dt  +  267)  log  v. 
Subtracting  equation  (2),  we  have 

dr  =  Nlogvdt.     ....     (3) 

Let  e  be  the  quantity  of  heat  employed  to  maintain 
the  temperature  of  the  gas  constant  during  its 


APPENDIX  B.  247 

dilatation.  According  to  the  reasoning  of  page  69, 
Sr  will  be  the  power  developed  by  the  fall  of  the 
quantity  e  of  heat  from  the  degree  t  -f-  td  to  the 
degree  t.  If  we  call  u  the  motive  power  developed 
by  the  fall  of  unity  of  heat  from  the  degree  t  to  the 
degree  zero,  as,  according  to  the  general  principle 
established  page  68,  this  quantity  u  ought  to  de- 
pend solely  on  i,  it  could  be  represented  by  the 
function  Ft,  whence  u  =  Ft. 

When  t  is  increased  it  becomes  t  +  td,  u  be- 
comes u  +  du  ;  whence 


Subtracting  the  preceding  equation,  we  have 
du  =  F(t  +  df)  -  Ft  =  F'tdt. 

This  is  evidently  the  quantity  of  motive  power 
produced  by  the  fall  of  unity  of  heat  from  the 
temperature  t  +  dt  to  the  temperature  t. 

If  the  quantity  of  heat  instead  of  being  a  unit 
had  been  e,  its  motive  power  produced  would  have 
had  for  its  value 

edu  =  eF'tdt  .....  (4) 
But  edu  is  the  same  thing  as  dr\  both  are  the 
power  developed  by  the  fall  of  the  quantity  e  of 
heat  from  the  temperature  t  -j-  dt  to  the  tempera- 
ture t;  consequently, 

edu  =  dr, 


248  APPENDIX  B. 

and  from  equations  (3),  (4), 

eF'tdt  =  N  \ogvdt; 
or,  dividing  by  F'tdt, 

N 
e=  -j^\ogv  =  Tlogv. 

JV 
Calling  T  the  fraction  -^  which  is  a  function  of  t 

only,  the  equation 

e  =  T  log  v 

is  the  analytical  expression  of  the  law  stated  pp.  80, 
81.  It  is  common  to  all  gases,  since  the  laws  ot 
which  we  have  made  use  are  common  to  all. 

If  we  call  s  the  quantity  of  heat  necessary  to 
change  the  air  that  we  have  employed  from  the 
volume  1  and  from  the  temperature  zero  to  the 
volume  v  and  to  the  temperature  t,  the  difference 
between  s  and  e  will  be  the  quantity  of  heat  re- 
quired to  bring  the  air  at  the  volume  1  from  zero 
to  t.  This  quantity  depends  on  t  alone;  we  will 
call  it  U.  It  will  be  any  function  whatever  of  t. 
We  shall  have 

s  =  e  +  U=  Tlogv  +  U. 

If  we  differentiate  this  equation  with  relation  to  t 
alone,  and  if  we  represent  it  by  T'  and  U',  the  dif- 
ferential coefficients  of  T  and  U,  we  shall  get 

//<? 

g=ZMogt;+Z7';     ...     (5) 


APPENDIX  B.  249 


-j2  is  simply  the  specific  heat  of  the  gas  under 
cl  t 

constant  volume,  and  our  equation  (1)  is  the  an- 
alytical expression  of  the  law  stated  on  page  86. 

If  we  suppose  the  specific  heat  constant  at  all 
temperatures  (hypothesis  discussed  above,  page  92), 

ds 

the  quantity  —  '-  will  be  independent  of  t',  and  in 
dt 

order  to  satisfy  equation  (5)  for  two  particular 
values  of  v,  it  will  be  necessary  that  T'  and  U'  be 
independent  of  t;  we  shall  then  have  T'  =  C,  a 
constant  quantity.  Multiplying  T'  and  C  by  dt, 
and  taking  the  integral  of  both,  we  find 


but  as  T  =  •=-  ,  we  have 


-  T    ~  Ct  +  C; 
Multiplying  both  by  dt  and  integrating,  we  have 

&  =  £  log  (01  +  C,)  +  C,; 

or  changing  arbitrary   constants,  and   remarking 
further  that  Ft  is  0  when  t  =  0°, 

Ft 
The  nature  of  the  function  Ft  would  be  thus 


=  A  log  (l  +  |)  .     .    .    .     (6) 


250  APPENDIX  B. 

determined,  and  we  would  thus  be  able  to  estimate 
the  motive  power  developed  by  any  fall  of  heat. 
But  this  latter  conclusion  is  founded  on  the  hy- 
pothesis of  the  constancy  of  the  specific  heat  of  a 
gas  which  does  not  change  in  volume  —  an  hypoth- 
esis which  has  not  yet  been  sufficiently  verified  by 
experiment.  Until  there  is  fresh  proof,  our  equa- 
tion (6)  can  be  admitted  only  throughout  a  limited 
portion  of  the  thermometric  scale. 

In  equation  (5),  the  first  member  represents,  as 
we  have  remarked,  the  specific  heat  of  the  air  oc- 
cupying the  volume  v.  Experiment  having  shown 
that  this  heat  varies  little  in  spite  of  the  quite  con- 
siderable changes  of  volume,  it  is  necessary  that 
the  coefficient  T'  of  log  v  should  be  a  very  small 
quantity.  If  we  consider  it  nothing,  and,  after 
having  multiplied  by  dt  the  equation 

Z"=0, 
we  take  the  integral  of  it,  we  find 

T=  C,    constant  quantity; 
but 


~  F't' 

whence 

_,,      N      N 

Ft  =  -?=-{?  =  A; 

whence  we  deduce  finally,  by  a  second  integration, 
Ft  =  At      B. 


APPENDIX  B.  251 

As  Ft  =  0  when  t  =  0,  B  is  0;  thus 


that  is,  the  motive  power  produced  would  be  found 
to  be  exactly  proportional  to  the  fall  of  the  caloric. 
This  is  the  analytical  translation  of  what  was 
stated  on  page  98. 

NOTE  F.  —  M.  Dalton  believed  that  he  had  dis- 
covered that  the  vapors  of  different  liquids  at  equal 
thermometric  distances  from  the  boiling-point 
possess  equal  tensions;  but  this  law  is  not  pre- 
cisely exact;  it  is  only  approximate.  It  is  the 
same  with  the  law  of  the  proportionality  of  the 
latent  heat  of  vapors  with  their  densities  (see  Ex- 
tracts from  a  Memoire  of  M.  C.  Despretz,  Annales 
de  CMmie  et  de  Physique,  t.  xvi.  p.  105,  and  t. 
xxiv.  p.  323).  Questions  of  this  nature  are  closely 
connected  with  those  of  the  motive  power  of  heat. 
Quite  recently  MM.  H.  Davy  and  Faraday,  after 
having  conducted  a  series  of  elegant  experiments 
on  the  liquefaction  of  gases  by  means  of  consider- 
able pressure,  have  tried  to  observe  the  changes  of 
tension  of  these  liquefied  gases  on  account  of  slight 
changes  of  temperature.  They  have  in  view  the 
application  of  the  new  liquids  to  the  production 
of  motive  power  (see  Annales  de  CMmie  et  de 
Physique,  January,  1824,  p.  80). 


252  APPENDIX  B. 

According  to  the  above-mentioned  theory,  we 
can  foresee  that  the  use  of  these  liquids  would 
present  no  advantages  relatively  to  the  economy 
of  heat.  The  advantages  would  be  found  only  in 
the  lower  temperature  at  which  it  would  be  possi- 
ble to  work,  and  in  the  sources  whence,  for  this 
reason,  it  would  become  possible  to  obtain  caloric. 

NOTE  G. — This  principle,  the  real  foundation 
of  the  theory  of  steam-engines,  was  very  clearly 
developed  by  M.  Clement  in  a  memoir  presented 
to  the  Academy  of  Sciences  several  years  ago. 
This  Memoir  has  never  been  printed,  and  I  owe 
the  knowledge  of  it  to  the  kindness  of  the  author. 
Not  only  is  the  principle  established  therein,  but 
it  is  applied  to  the  different  systems  of  steam- 
engines  actually  in  use.  The  motive  power  of 
each  of  them  is  estimated  therein  by  the  aid  of 
the  law  cited  page  92,  and  compared  with  the  re- 
sults of  experiment. 

The  principle  in  question  is  so  little  known  or 
so  poorly  appreciated,  that  recently  Mr.  Perkins,  a 
celebrated  mechanician  of  London,  constructed  a 
machine  in  which  steam  produced  under  the  pres- 
sure of  35  atmospheres — a  pressure  never  before 
used — is  subjected  to  very  little  expansion  of  vol- 
ume, as  any  one  with  the  least  knowledge  of  this 
machine  can  understand.  It  consists  of  a  single 
cylinder  of  very  small  dimensions,  which  at  ench 


APPENDIX  B.  253 

stroke  is  entirely  filled  with  steam,  formed  under 
the  pressure  of  35  atmospheres.  The  steam  pro- 
duces no  effect  by  the  expansion  of  its  volume,,  for 
no  space  is  provided  in  which  the  expansion  can 
take  place.  It  is  condensed  as  soon  as  it  leaves 
the  small  cylinder.  It  works  therefore  only  under 
a  pressure  of  35  atmospheres,  and  not,  as  its  use- 
ful employment  would  require,  under  progressively 
decreasing  pressures.  The  machine  of  Mr.  Per- 
kins seems  not  to  realize  the  hopes  which  it  at 
first  awakened.  It  has  been  asserted  that  the 
economy  of  coal  in  this  engine  was  j\  above  the 
best  engines  of  Watt,  and  that  it  possessed  still 
other  advantages  (see  Annales  de  Chimie  et  de 
Physique,  April,  1823,  p.  429).  These  assertions 
have  not  been  verified.  The  engine  of  Mr.  Per- 
kins is  nevertheless  a  valuable  invention,  in  that 
it  has  proved  the  possibility  of  making  use  of 
steam  under  much  higher  pressure  than  previously, 
and  because,  being  easily  modified,  it  may  lead  to 
very  useful  results. 

Watt,  to  whom  we  owe  almost  all  the  great  im- 
provements in  steam-engines,  and  who  brought 
these  engines  to  a  state  of  perfection  difficult 
even  now  to  surpass,  was  also  the  first  who  em- 
ployed steam  under  progressively  decreasing  pres- 
sures. In  many  cases  he  suppressed  the  introduc- 
tion of  the  steam  into  the  cylinder  at  a  half,  a 


254 


APPENDIX  B. 


third,  or  a  quarter  of  the  stroke.  The  piston  com- 
pletes its  stroke,  therefore,  under  a  constantly 
diminishing  pressure.  The  first  engines  working  on 
this  principle  date  from  1778.  Watt  conceived  the 
idea  of  them  in  1769,  and  took  out  a  patent  in  1782. 
We  give  here  the  Table  appended  to  Watt's 
patent.  He  supposed  the  steam  introduced  into 
the  cylinder  during  the  first  quarter  of  the  stroke  of 
the  piston;  then,  dividing  this  stroke  into  twenty 
parts,  he  calculated  the  mean  pressure  as  follows: 


Portions  of  the  descent  from  the 
top  of  the  cylinder. 


Decreasing  pressure  of  the 
steam,  the  entire  pressure 
being  1. 


0.05 

f  1.0001 

0.10 
0.15 

Steam  arriving     1.000  |   T      , 
-     freely    from  ^  1.000  [  Lo™L    l 

0.20 

the  boiler. 

1.000 

Quarter.  .  . 

..0.25 

,1.000  J 

0.30 

r  0.830 

0.35 

0.714 

0.40 

0.625 

Half  ... 

0.45 
0.50 
0  55 

The  steam  be- 

5'Sgj Half  original 
0454^       pressure. 

o.'eo 

0.65 
0.70 
0.75 

0.80 

ing   cut    off 
and  the    de- 
scent  taking 
place  only  by 
expansion. 

0'.417 
0.385 
0.375 
0.333     One  third. 
0.312 

0.85 

0.294 

0.90 

0.277 

Bottom  of 

0.95 

0.262 

cylinder. 

..1.00 

[0.025     Quarter. 

Total,  11.583 

11  583 
Mean  pressure  -       --  =  0.579. 


APPENDIX  R  255 

On  which  he  remarked,,  that  the  mean  pressure  is 
more  than  half  the  original  pressure;  also  that  in 
employing  a  quantity  of  steam  equal  to  a  quarter, 
it  would  produce  an  effect  more  than  half, 

Watt  here  supposed  that  steam  follows  in  its  ex- 
pansion the  law  of  Mariotte,  which  should  not  be 
considered  exact,  because,  in  the  first  place,  the 
elastic  fluid  in  dilating  falls  in  temperature,  and 
in  the  second  plac3  there  is  nothing  to  prove  that 
a  part  of  this  fluid  is  not  condensed  by  its  expan- 
sion. Watt  should  also  have  taken  into  considera- 
tion the  force  necessary  to  expel  the  steam  which 
remains  after  condensation,  and  which  is  found  in 
quantity  as  much  greater  as  the  expansion  of  the 
volume  has  been  carried  further.  Dr.  Robinson 
has  supplemented  the  work  of  Watt  by  a  simple 
formula  to  calculate  the  effect  of  the  expansion  of 
steam,  but  this  formula  is  found  to  have  the  same 
faults  that  we  have  just  noticed.  It  has  neverthe- 
less been  useful  to  constructors  by  furnishing  them 
approximate  data  practically  quite  satisfactory. 
We  have  considered  it  useful  to  recall  these  facts 
because  they  are  little  known,  especially  in 
France.  These  engines  have  been  built  after  the 
models  of  the  inventors,  but  the  ideas  by  which 
the  inventors  were  originally  influenced  have  been 
but  little  understood.  Ignorance  of  these  ideas 


256  APPENDIX  B. 

has  often  led  to  grave  errors.  Engines  originally 
well  conceived  have  deteriorated  in  the  hands  of 
unskilful'  builders,  who,  wishing  to  introduce  in 
them  improvements  of  little  value,  have  neglected 
the  capital  considerations  which  they  did  not  know 
enough  to  appreciate. 

NOTE  H. — The  advantage  in  substituting  two 
cylinders  for  one  is  evident.  In  a  single  cylinder 
the  impulsion  of  the  piston  would  be  extremely 
variable  from  the  beginning  to  the  end  of  the 
stroke.  It  would  be  necessary  for  all  the  parts  by 
which  the  motion  is  transmitted  to  be  of  sufficient 
strength  to  resist  the  first  impulsion,  and  perfectly 
fitted  to  avoid  the  abrupt  movements  which  would 
greatly  injure  and  soon  destroy  them.  It  would 
be  especially  on  the  working  beam,  on  the 
supports,  on  the  crank,  on  the  connecting-rod, 
and  on  the  first  gear-wheels  that  the  unequal 
effort  would  be  felt,  and  would  produce  the 
most  injurious  effects.  It  would  be  necessary 
that  the  steam-cylinder  should  be  both  sufficiently 
strong  to  sustain  the  highest  pressure,  and  with 
a  large  enough  capacity  to  contain  the  steam 
after  its  expansion  of  volume,  while  in  using  two 
successive  cylinders  it  is  only  necessary  to  have 
the  first  sufficiently  strong  and  of  medium  ca- 
pacity,— which  is  not  at  all  difficult, — and  to  have 


APPENDIX  B.  257 

the  second  of  ample  dimensions,  with  moderate 
strength. 

Double-cylinder  engines,  although  founded  on 
correct  principles,  often  fail  to  secure  the  advan- 
tages expected  from  them.  This  is  due  principally 
to  the  fact  that  the  dimensions  of  the  different 
parts  of  these  engines  are  difficult  to  adjust,  and 
that  they  are  rarely  found  to  be  in  correct  propor- 
tion. Good  models  for  the  construction  of  double- 
cylinder  engines  are  wanting,  while  excellent  de- 
signs exist  for  the  construction  of  engines  on  the 
plan  of  Watt.  From  this  arises  the  diversity  that 
we  see  in  the  results  of  the  former,  and  the  great 
uniformity  that  we  have  observed  in  the  results  of 
the  latter. 

NOTE  I. — Among  the  attempts  made  to  develop  \ 
the  motive  power  of  heat  by  means  of  atmospheric 
air,  we  should  mention  those  of  MM.  Niepce,  which 
were  made  in  France  several  years  ago,  by  means 
of  an  apparatus  called  by  the  inventors  a  pyre- 
olophore.  The  apparatus  was  made  thus:  There 
was  a  cylinder  furnished  with  a  piston,  into  which 
the  atmospheric  air  was  introduced  at  ordinary 
density.  A  very  combustible  material,  reduced  to 
a  condition  of  extreme  tenuity,  was  thrown  into  it, 
remained  a  moment  in  suspension  in  the  air,  and 
then  flame  was  applied.  The  inflammation  pro- 


258  APPENDIX  B. 

duced  very  nearly  the  same  effect  as  if  the  elastic, 
fluid  had  been  a  mixture  of  air  and  combustible 
gas,  of  air  and  carburetted  hydrogen  gas,,  for  ex- 
ample. There  was  a  sort  of  explosion,  and  a  sud- 
den dilatation  of  the  elastic  fluid — &  dilatation  that 
was  utilized  by  making  it  act  upon  the  piston. 
The  latter  may  have  a  motion  of  any  amplitude 
whatever,  and  the  motive  power  is  thus  realized. 
The  air  is  next  renewed,  and  the  operation  re- 
peated. 

This  machine,  very  ingenious  and  interesting, 
especially  on  account  of  the  novelty  of  its  princi- 
ple, fails  in  an  essential  point.  The  material  used 
as  a  combustible  (it  was  the  dust  of  Lycopodium, 
used  to  produce  flame  in  our  theatres)  was  so  ex- 
pensive, that  all  the  advantage  was  lost  through 
that  cause;  and  unfortunately  it  was  difficult  to 
employ  a  combustible  of  moderate  price,  since  a 
very  finely  powdered  substance  was  required  which 
would  burn  quickly,  spread  rapidly,  and  leave  little 
or  no  ash. 

Instead  of  working  as  did  MM.  Niepce,  it  would 
seem  to  us  preferable  to  compress  the  air  by  means 
of  pumps,  to  make  it  traverse  a  perfectly  closed 
furnace  into  which  the  combustible  had  been  in- 
troduced in  small  portions  by  a  mechanism  easy  of 
conception,  to  make  it  develop  its  action  in  a  cylin- 


APPENDIX  B.  259 

der  with  a  piston,  or  in  any  other  variable  space; 
finally,  to  throw  it  out  again  into  the  atmosphere, 
or  even  to  make  it  pass  under  a  steam-boiler  in 
order  to  utilize  the  temperature  remaining. 

The  principal  difficulties  that  we  should  meet  in 
this  mode  of  operation  would  be  to  enclose  the  fur- 
nace in  a  sufficiently  strong  envelope,  to  keep  the 
combustion  meanwhile  in  the  requisite  state,  to 
maintain  the  different  parts  of  the  apparatus  at  a 
moderate  temperature,  and  to  prevent  rapid  abra- 
sion of  the  cylinder  and  of  the  piston.  These  dif- 
ficulties do  not  appear  to  be  insurmountable. 

There  have  been  made,  it  is  said,  recently  in 
England,  successful  attempts  to  develop  motive 
power  through  the  action  of  heat  on  atmospheric 
air.  We  are  entirely  ignorant  in  what  these  at- 
tempts have  consisted — if  indeed  they  have  really 
been  made. 

NOTE  J. — The  result  given  here  was  furnished  by 
an  engine  whose  large  cylinder  was  45  inches  in 
diameter  and  7  feet  stroke.  It  is  used  in  one  of  the 
mines  of  Cornwall  called  Wheal  Abraham.  This 
result  should  be  considered  as  somewhat  excep- 
tional, for  it  was  only  temporary,  continuing  but  a 
single  month.  Thirty  millions  of  Ibs.  raised  one 
English  foot  per  bushel  of  coal  of  88  Ibs.  is  generally 
regarded  as  an  excellent  result  for  steam-engines. 


260  APPENDIX  B. 

It  is  sometimes  attained  by  engines  of  the  Watt 
type,  but  very  rarely  surpassed.  This  latter  prod- 
uct amounts,  in  French  measures,  to  104,000  kilo- 
grams raised  one  metre  per  kilogram  of  coal  con- 
sumed. 

According  to  what  is  generally  understood  by 
one  horse-power,  in  estimating  the  duty  of  steam- 
engines,  an  engine  of  ten  horse-power  should  raise 
per  second  10  X  75  kilograms,  or  750  kilograms,  to 
a  height  of  one  metre,  or  more,  per  hour;  750  X 
3600  =  2,700,000  kilograms  to  one  metre.  If  we 
suppose  that  each  kilogram  of  coal  raised  to  this 
height  104,000  kilograms,  it  will  be  necessary,  in 
order  to  ascertain  how  much  coal  is  burnt  in  one 
hour  by  our  ten-horse-power  engine,  to  divide 
2,700,000  by  104,000,  which  gives  *fiff-  =  26  kilo- 
grams. Now  it  is  seldom  that  a  ten -horse-power 
engine  consumes  less  than  26  kilograms  of  coal  per 
hour. 


APPENDIX  C. 

NOTE  BY  THE  EDITOR. 

ALL  the  preceding  data  are  to-day  subject  to 
modification. 

Thus  a  duty  of  150,000,000  ft.-lbs.  per  100  Ibs. 
good  coal  is  to-day  attainable,  and  two  thirds  that 
figure  is  extremely  common.  With  engines  of 
large  size  the  coal-consumption  has  fallen  to  one 
half,  sometimes  even  to  one  fourth,  the  figure  in 
the  text. 

Hot-air  engines  are  superseded  by  the  gas- 
engine  and  the  oil-vapor  engine ;  which  even 
threaten,  in  the  opinion  of  many  engineers,  to 
ultimately  displace  the  steam-engine. 

Compound  and  other  multiple-cylinder  engines, 
with  two,  three,  and  even  four  cylinders  in  series, 
are  now  always  employed  where  fuel  is  costly.  The 
reason  of  their  success  is,  in  part,  that  given  in 
Note  H;  but  in  only  small  part.  The  real  cause 
of  their  general  adoption  is  the  fact  that  the  in- 
ternal thermal  waste  by  "cylinder-condensation" 
— which  in  simple  engines  ordinarily  amounts, 
according  to  size,  to  from  25  to  50  per  cent,  or 

261 


262  APPENDIX  C. 

more,  of  all  heat  supplied  by  the  boiler — is  reduced 
nearly  in  proportion  to  the  number  of  steam -cylin- 
ders in  series. 

For  the  applied  thermodynamics  of  the  steam- 
engine,  following  Carnot  and  Thomson,  see  the 
pages  of  Kankine  and  of  Clausius  of  1850  to  1860, 
and  especially  the  treatise  of  Rankine  on  the 
Steam-engine.  The  editor  has  adopted  the  methods 
of  these  great  successors  of  Carnot  in  his  "  Manual 
of  the  Steam-engine"  (2  vols.  8vo;  N.  Y.,  J.  Wiley 
&  Sons),  which  may  be  consulted  in  this  connec- 
tion, and  especially  for  details  of  the  theory  and 
the  structure  of  this  prime  mover. 


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